# A classification of asymptotic behaviors of Green functions of random walks in the quadrant

Irina Ignatiouk-Robert

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Irina Ignatiouk-Robert. A classification of asymptotic behaviors of Green functions of random walks in the quadrant. Electronic Journal of Probability, 2025, 30, 10.1214/24-EJP1252. hal-05337174

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Electron. J. Probab. 30 (2025), article no. (non), 1–104.
ISSN: 1083-6489 https://doi.org/10.1214/24-EJP1252

# A Classification of Asymptotic Behaviors of Green Functions of Random Walks in the Quadrant

Irina Ignatiouk-Robert *

## Abstract

This paper investigates the asymptotic behavior of Green functions associated to partially homogeneous random walks in the quadrant $Z_+^2$. There are four possible distributions for the jumps of these processes, depending on the location of the starting point: in the interior, on the two positive axes of the boundary, and at the origin (0, 0).

With mild conditions on the positive jumps of the random walk, which can be unbounded, a complete analysis of the asymptotic behavior of the Green function of the random walk killed at (0, 0) is achieved. The main result is that *eight* regions of the set of parameters determine completely the possible limiting behaviors of Green functions of these Markov chains. These regions are defined by a set of relations for several characteristics of the distributions of the jumps.

In the transient case, a description of the Martin boundary is obtained and in the positive recurrent case, our results give the exact limiting behavior of the invariant distribution of a state whose norm goes to infinity along some asymptotic direction in the quadrant. These limit theorems extend results of the literature obtained, up to now, essentially for random walks whose jump sizes are either 0 or 1 on each coordinate.

Our approach relies on a combination of several methods: probabilistic representations of solutions of analytical equations, Lyapounov functions, convex analysis, methods of homogeneous random walks, and complex analysis arguments.

**Keywords:** Green Functions; Martin Boundary; Random Walks.

**MSC2020 subject classifications:** 60J45;31C05;60J10;60J50.

Submitted to EJP on May 9, 2024, final version accepted on December 3, 2024.

Supersedes arXiv:2311.06652.

<footnote>*AGM-UMR8088, CNRS CY, Département de mathématiques, CY Cergy Paris Université, 2 Avenue Adolphe Chauvin
95302 Cergy-Pontoise Cedex, France
E-mail: Irina.Ignatiouk@u-cergy.fr</footnote>
# 1 Introduction

For a transient Markov chain $(Z(n))$ on an infinite countable state space $\mathcal{E}$ determining all possible limits of the associated Martin kernel, the Martin boundary, is an important and difficult problem in general. By the Poisson-Martin representation theorem, it gives the Martin compactification of the state space, an integral representation of all non-negative harmonic functions. For an introduction to the theory of Martin boundary for countable Markov chains, see the classical references Doob [5] and Dynkin [6].

The characterization of the Martin boundary for homogeneous random walks in $\mathbb{Z}^d$ has been obtained in Ney and Spitzer [27], via a set of technical estimates related to the local central limit theorem. The Martin boundary has been also identified for random walks on free groups, hyperbolic graphs and Cartesian products. See Woess [30] for a thorough presentation of boundary theory of random walks.

There are few results for more general Markov chains. For random walks on non-homogeneous trees, the Martin boundary has been obtained in Cartier [3]. Doney [4] identified the Martin boundary of homogeneous random walks $(Z(n))$ on $\mathbb{Z}$ killed on the negative half-line of $\mathbb{Z}$. For space-time random walks $(S(n))=(n, Z(n))$ associated to a homogeneous random walk $Z(n)$ on $\mathbb{Z}$ killed on the negative half-line, the Martin boundary is obtained in Alili and Doney [1]. The proof of these results relies on the one-dimensional structure of these processes.

The Martin boundary for partially homogeneous random walks killed or reflected on a half-space or a cone of $\mathbb{Z}^d$ has been identified with large deviation techniques, Choquet-Deny theory and ratio limit theorems of Markov-additive processes. See Ignatiouk [15, 16, 17].

## Random Walks on $\mathbb{Z}_+^2$

In this paper we consider a partially homogeneous random walk $(Z(n))$ on $\mathbb{Z}_+^2$ with the following characteristics: the distribution of its jumps is

a) $\mu$ in the interior of $\mathbb{Z}_+^2$;

b) $\mu_0$ at $0=(0,0)$;

c) $\mu_1$ in $\{0\} \times (\mathbb{Z}_+ \setminus \{0\})$;

d) $\mu_2$ in $(\mathbb{Z}_+ \setminus \{0\}) \times \{0\}$;

The possible negative jumps are either 0 or $-1$ on each coordinate for $\mu, \mu_1$ and $\mu_2$.

When it is transient, the Green function $G$ of the Markov chain is, for $j, k \in \mathbb{Z}_+^2$,

$$G(j, k) \stackrel{\text{def.}}{=} \sum_{n=0}^{+\infty} \mathbb{P}_j(Z(n)=k).$$

The strict hitting time of $0=(0,0)$ is denoted as

$$\tau_0 \stackrel{\text{def.}}{=} \inf\{k>0 : Z(k)=(0,0)\}$$

and the Green function of the Markov chain killed at 0 is, for $j \in \mathbb{Z}_+^2$, $k \in \mathbb{Z}_+^2 \setminus \{0\}$,

$$g(j, k) \stackrel{\text{def.}}{=} \sum_{n \ge 0} \mathbb{P}_j(Z(n) = k, n < \tau_0).$$

## Exact Asymptotics of Green Functions

The Martin Kernel being the ratio of two Green functions, its limiting behavior can be obtained from the exact asymptotics of $G(j, k)$ when $k$ goes to infinity. Ney and Spitzer [27] determines the Martin boundary of homogeneous random walks in $\mathbb{Z}^d$ in this way.

It is easily seen that, for $j, k \in \mathbb{Z}_+^2 \setminus \{0\}$,

$$G(j, k) = g(j, k) + G(j, 0)g(0, k) \text{ and } G(0, k) = G(0, 0)g(0, k),$$
which gives a relation between the asymptotic behaviors of the Green functions $k \mapsto G(j, k)$ and $k \mapsto g(j, k)$.

This type of asymptotic analysis can also be used to investigate positive recurrent Markov chains. The invariant distribution $\pi$ is represented with the Green function of the Markov chain killed at 0, for $k \in \mathbb{Z}_+^2$,

$$ \pi(k) = \pi(0) \sum_{n=0}^{+\infty} \mathbb{P}_0(Z(n)=k, n<\tau_0) = \pi(0)g(0, k), \quad (1.1) $$

The limiting behavior of $k \mapsto g(0, k)$ gives therefore the asymptotic behavior of the invariant distribution of a state $k$ going to infinity.

## Convergence to Infinity

For $j \in \mathbb{Z}_+^2$, we will investigate the asymptotic behavior of $g(j, k)$ as $k=(k_1, k_2) \in \mathbb{Z}_+^2$ gets large in several ways.

(1) With direction $u \in S_+^1 = \{x=(x_1, x_2) \in \mathbb{R}_+^2 : \|x\|=x_1^2+x_2^2=1\}$,

$$ \min(k_1, k_2) \to +\infty, \quad \text{and} \quad \frac{k}{\|k\|} \to u. $$

(2) Along the axes.

The quantity $k_2 \in \mathbb{Z}_+$ varies in a finite subset of $\mathbb{Z}_+$ and $k_1 \to +\infty$, and the symmetrical case by exchanging the variables $k_1$ and $k_2$.

The case (1) is the classical set of asymptotic behaviors considered in general. As we will see, the asymptotics of the Green function for the case (2) are different from the case (1), they depend on $k_2$ and exhibit interesting behaviors. They have also been considered in Kobayashi and Miyazawa [20] for random walks with jumps of size 1. Note that there is a slight abuse of terminology for (2) since, strictly speaking, the cases $u=(1, 0)$ and $u=(0, 1)$ of (1) are also “along the axes”.

## A Functional Equation

A functional equation for generating functions of the Green function of the Markov chain killed at 0 plays a central role in our study. It is expressed as, for $j \in \mathbb{Z}_+^2$ and $(x, y)$ in a convenient subset of $\mathbb{C}^2$,

$$ (1-P(x, y))(H_j(x, y)-H_j(x, 0)-H_j(0, y)) = L_j(x, y) + (\phi_1(x, y)-1)H_j(x, 0) + (\phi_2(x, y)-1)H_j(0, y) \quad (1.2) $$

holds, where the quantity $(L_j(x, y))$ is a known function (defined by (2.27)) and

(1) $(H_j(x, y))$ is the generating function of $(g(j, k), k \in \mathbb{Z}_+^2)$;

(2) $P(x, y)$ is the generating function of $\mu$ at $(x, y)$, the distribution of the jumps in the interior of $\mathbb{Z}_+^2$. The quantity $Q(x, y)=xy(1-P(x, y))$ is in general referred to as the kernel;

(3) For $i=1, 2$, $\phi_i(x, y)$ is the generating function of $\mu_i$ at $(x, y)$.

(See Theorem 2.1 and its proof in Section 6).

The Kernel method consists in finding a convenient root $Y(x)$ of the equation $Q(x, y) = 0$ in order to inject it to the functional equation (1.2). The left hand side of the resulting equation
will became then zero, and one can use its right hand side in order to investigate the analytical
properties of the function $H_j(x, 0)$.

In most of studies on the asymptotic behavior of Green functions or invariant measures of
random walks in the quadrant, there is always an associated functional equation similar to
equation (1.2).

## Literature

We now give a brief presentation of the existing asymptotic results for non-homogeneous
random walks in the quadrant.

(1) Nearest neighbor random walks: Jumps of size 0 or 1 on each coordinate.

In this case, the kernel $Q(x, y)$ is a polynomial of degree at most 2 in each variable and
hence for any $x \in \mathbb{C}$, the kernel equation $Q(x, y) = 0$ has exactly two roots (counting
their multiplicity) having an explicit form. In the positive recurrent case, by using
methods of complex analysis on elliptic curves, the asymptotic behavior of the invariant
distribution along lines of $\mathbb{Z}_+^2$ has been obtained in the early work Malyshev [25] in 1973.
Following these ideas, extensions of these results have been established in Kurkova and
Malyshev [22], Kurkova and Raschel [23], Raschel [28] and Li and Zhao [24], but, as in
the original paper [25], only when positive jumps are of size 1. See Fayolle et al. [8] for a
general presentation and additional references therein.

In the positive recurrent case, Kobayashi and Miyazawa [20] determines the exact
domain of convergence of the generating function of stationary distribution and obtains
the asymptotic behavior of the stationary distribution.

In both cases, [20] and [25], the analysis relies in an essential way on the explicit
representation of the roots of the quadratic kernel equation $Q(x, y) = 0$. This is the main
limitation of this type of approach.

(2) Reflected Brownian motion. For reflected Brownian motion in $\mathbb{R}_+^2$, similar problems and with an analogous approach to Malyshev [25] were investigated in a series of papers Franceschi and Kourkova [12], Franceschi [11], Franceschi et al.[13] . In this case, the kernel equation is also quadratic and its roots are explicit.

(3) For positive recurrent random walks with unbounded positive jumps, exact asymptotics of the stationary measure have been investigated in Borovkov and Mogulskii [2] with large deviation techniques. The asymptotics are considered for interior directions of $S_+^1$. Some constants do not seem to be explicitly determined in the limit results of this reference. In particular, it is not clear how the limiting behavior of the invariant distribution of a state going to infinity depends on the asymptotic direction $u \in S_+^1$.

(4) Positive recurrent random walks with unbounded positive jumps have also been analyzed in Kobayashi and Miyazawa [21] from the point of view of tail asymptotics. For these asymptotics, a line of $\mathbb{Z}_+^2$ associated to a fixed vector is going to infinity in the sense that its distance to $(0, 0)$ is going to infinity. The quantity considered for the tail asymptotics is the invariant distribution of all states of $\mathbb{Z}_+^2$ above this line. The exact domain of convergence of the generating function of the invariant distribution is obtained and, with methods of Markov-additive processes, exact tail asymptotics with explicit constants are derived.

(5) Ignatiouk et al. [19] has investigated transient random walks in the quadrant $\mathbb{Z}_+^2$ with unbounded positive jumps, the size of negative jumps is not necessarily 1 but bounded. An additional assumption in [19] is that the random walk escapes to the infinity along the
horizontal axis and the vertical axis. With methods of local Markov-additive processes and complex analysis arguments, the exact asymptotics of the Green function are obtained. They are expressed in terms of asymptotics of Green functions of random walks in half-plane, i.e. with one boundary removed. Our present paper shows that such a result is wrong in general, both boundaries play a role in fact in several of our convergence results.

(6) Additionally, in Viet Hung Hoang and al. [14], the authors have obtained exact asymptotics of the Green function for a singular random walk by using specific properties of their model. In Ignatiouk-Robert [18], exact asymptotics of the Green function have been obtained for a homogeneous random walk in $Z^d$ killed outside of an open cone by using methods of functional equations, integral representations of the Green function and Woess' approach for the case of homogeneous random walks in $Z^d$. The technical results of the last paper are used in the present paper in order to get the asymptotic behavior of the Green function for the directions in the interior of the quadrant.

## A Quick Presentation

A significant part of our paper is devoted to the definition and the properties of the classification in eight regions of the space of parameters, see Proposition 2.2 below. It is defined as a set of relations for several characteristics associated to the distributions of the jumps $\mu, \mu_i, i \in \{1, 2\}$. For each region of this classification, an investigation of the analyticity properties of the generating functions functions $H_j, j \in Z_2^+$ and the study of the nature of their dominant singularities are achieved. With these results, the exact asymptotics of the Green function $(g(j, k), j, k \in \mathbb{N}^2)$ are derived. They are stated in Theorems 2.4 and 2.6 of Section 2.

In the literature, asymptotic results for nearest neighbor random walks in the quadrant are often formulated either under conditions of positive recurrence, see [8, 24, 25, 26], or under conditions of transience, see [22], and use in both cases an explicit form of the roots of the quadratic equation $Q(x, y) = 0$. In [8, 24, 25, 22] results are obtained, a kind of classification of asymptotic behaviors, separately for each configuration of the mean drifts. In [26] the authors have considered only positive recurrent random walks and the classification of asymptotic behaviors of stationary probabilities is given in terms of the domain of convergence of the generating function of the stationary probabilities of the process. If the classification of this last reference has some similarities with our classification, it still use the conditions of positive recurrence, formulated in terms of the mean drifts. In our paper, we give a common classification in eight regions of the space of parameters which is valid both for positive recurrent and transient random walks and is not determined by the mean drifts but by the mutual disposition of the level sets of the jump generating functions.

We now give a sketch of the general method used to obtain these convergence results. Section 2 gives a much more detailed description of the contributions and also of the methods used.

In the first step we study the solutions $(x, y)$ of the equation $Q(x, y)=xy(1-P(x, y))=0$. This is done by investigating the existence of a function $Y(x)$ defined on a subset of $\mathbb{C}$ such that $(x, Y(x))$ is a zero of $Q$. By canceling the left-hand side of (1.2), it gives a relation between $(H_j(x, 0))$ and $(H_j(0, Y(x)))$. An analogue study is done, by exchanging the roles of $x$ and $y$, with $(X(y), y)$. After this step, an analytic continuation of $(H_j(0, Y(x)))$ is achieved and, with relation (1.2) it gives an analytic continuation of $(H_j(x, 0))$. This is a difficult point and this is where a convenient representation of $(Y(x))$ is crucial. In our approach, the key argument in this step is a probabilistic representation of the function $(Y(x))$. Under general conditions, an
expression of the functions $(H_j(0, y))$ and $(H_j(x, 0))$ is derived, and therefore an expression of $(H_j(x, y))$, the generating function of $(g(j, k))$. The last step uses this representation and, with complex analysis arguments, one can derive the asymptotic behavior of $(g(j, k))$ when $k$ goes to infinity.

# 2 Overview of the Results

## General Notations

Throughout the paper, the following notations are used.

(1) For two points $(x_1, y_2)$, $(x_2, y_2) \in \mathbb{R}^2$, the line segment in $\mathbb{R}^2$ with the end-points $(x_1, y_2)$ and $(x_2, y_2)$ is denoted by $[(x_1, y_2), (x_2, y_2)]$. The unit circle of $\mathbb{R}_+^2$ is

$$S_+^1 \stackrel{\text{def.}}{=} \{w=(u, v) \in \mathbb{R}_+^2 : \|x\| = u^2 + v^2 = 1\}. \qquad (2.1)$$

(2) For $a \in \mathbb{C}$ and $r>0$, we denote by $B(a, r)$ the open disk in $\mathbb{C}$ with center $a$ and radius $r$. A poly-disc of $\mathbb{C}^2$ is the product of two open discs.

(3) For $r_2>r_1>0$, we let

$$C(r_1, r_2) = \{x \in \mathbb{C} : r_1 < |x| < r_2\} \quad \text{and} \quad \bar{C}(r_1, r_2) = \{x \in \mathbb{C} : r_1 \le |x| \le r_2\}.$$

(4) For a subset $B$ of $[0, +x\infty]$, we denote

$$\Omega(B) \stackrel{\text{def.}}{=} \{(x, y) \in \mathbb{C}^2 : (|x|, |y|) \in B\}. \qquad (2.2)$$

A set $B \subset [0, +\infty]$ is logarithmically convex (resp. strictly logarithmically convex) if, for any $x, y \in B$ and $\lambda \in [0, 1]$, $x^\lambda y^{1-\lambda} \in B$, resp. $x^\lambda y^{1-\lambda} \in \mathring{B}$ when $x \neq y$. We denote by LogCH($B$) the logarithmic convex hull of $B$ in $[0, +\infty]$, i.e. the smallest logarithmically convex set of $[0, +\infty]$ containing $B$.

To simplify some expressions, we will also use the notations, for a $C_2$-function $f$ on $\mathbb{C}^2$,

$$\partial_x f(x, y) = \frac{\partial f}{\partial x}(x, y), \quad \partial_y f(x, y) = \frac{\partial f}{\partial y}(x, y),$$

$$\partial_{xx}^2 f(x, y) = \frac{\partial^2 f}{\partial x^2}(x, y), \quad \partial_{xy}^2 f(x, y) = \frac{\partial^2 f}{\partial x \partial y}(x, y), \quad \partial_{yy}^2 f(x, y) = \frac{\partial^2 f}{\partial y^2}(x, y).$$

We now introduce the non-homogeneous random walks investigated in this paper.

## 2.1 Non-Homogeneous Random walks in $\mathbb{Z}_+^2$: Definitions and Assumptions

The process $Z(n)=(Z_1(n), Z_2(n))$ on $\mathbb{Z}_+^2$ is a Markov chain on $\mathbb{Z}_+^2$ with transition probabilities given for $j=(j_1, j_2) \in \mathbb{Z}_+^2$ by

$$\mathbb{P}_j(Z(1) = j+k) \stackrel{\text{def.}}{=} \begin{cases} \mu(k) & \text{if } j_1 > 0 \text{ and } j_2 > 0, \\ \mu_1(k) & \text{if } j_1 > 0 \text{ and } j_2 = 0, \\ \mu_2(k) & \text{if } j_2 > 0 \text{ and } j_1 = 0, \\ \mu_0(k) & \text{if } (j_1, j_2) = (0, 0), \end{cases} \qquad (2.3)$$

where $\mu$ is a probability measure on $\mathbb{Z}^2$, and $\mu_1, \mu_2$ and $\mu_0$ are sub-probability measures (positive measures with total mass less or equal than 1) on, respectively, $\mathbb{Z} \times \mathbb{Z}_+$, $\mathbb{Z}_+ \times \mathbb{Z}$ and $\mathbb{Z}_+^2$.
Their generating functions, defined on their set of convergence in $\mathbb{C}^2$, are denoted by

$$P(x, y) \stackrel{\text{def.}}{=} \sum_{j=(j_1, j_2) \in \mathbb{Z}^2} \mu(j) x^{j_1} y^{j_2}, \qquad (2.4)$$

$$\phi_1(x, y) \stackrel{\text{def.}}{=} \sum_{j=(j_1, j_2) \in \mathbb{Z}^2} \mu_1(j) x^{j_1} y^{j_2}, \quad \phi_2(x, y) \stackrel{\text{def.}}{=} \sum_{j=(j_1, j_2) \in \mathbb{Z}^2} \mu_2(j) x^{j_1} y^{j_2}, \qquad (2.5)$$

$$\phi_0(x, y) \stackrel{\text{def.}}{=} \sum_{j=(j_1, j_2) \in \mathbb{Z}^2} \mu_0(j) x^{j_1} y^{j_2}. \qquad (2.6)$$

The level sets $D, D_1$ and $D_2$ of these generating functions are defined as

$$D \stackrel{\text{def.}}{=} \{(x, y) \in ]0, +\infty[ : P(x, y) \le 1\} \quad \text{and} \quad D_i \stackrel{\text{def.}}{=} \{(x, y) \in ]0, +\infty[ : \phi_i(x, y) \le 1\}, \quad i \in \{1, 2\}. \qquad (2.7)$$

There are three main assumptions used in our results.

**Assumption (A1)**

(i) The homogeneous random walk associated to the distribution $\mu$ is irreducible on $\mathbb{Z}^2$;

(ii) The generating function $P$ is finite in a neighborhood of the set $D$ in $\mathbb{R}^2$;

(iii) The set $D$ has a non-empty interior.

**Assumption (A2)**

(i) For $j=(j_1, j_2) \in \mathbb{Z}^2$, $\mu(j_1, j_2)=0$ if $j_1<-1$ or $j_2<-1$.

**Assumption (A3)**

(i) The random walk $Z(n)=(Z_1(n), Z_2(n))$ is irreducible on $\mathbb{Z}_+^2$;

(ii) The generating functions $\phi_1, \phi_2$ and $\phi_0$ are finite in a neighborhood of the set $D$;

(iii) The sets $D \cap D_1$ and $D \cap D_2$ have a non-empty interior;

(iv) For $j=(j_1, j_2) \in \mathbb{Z}^2$,

- $\mu_1(j_1, j_2)=0$ if $j_1<-1$;

- $\mu_2(j_1, j_2)=0$ if $j_2<-1$;

(v) There exists $j=(j_1, j_2) \in \mathbb{Z}^2$ with $j_2>0$ such that $\mu_1(j_1, j_2)>0$;

(vi) There exists $j=(j_1, j_2) \in \mathbb{Z}^2$ with $j_1>0$ such that $\mu_2(j_1, j_2)>0$.

The Markov chain $(Z(n))$ killed at 0 and its Green function are now introduced.

**Definition 1 (Killed Markov Chain).** The return time of the process $(Z(n))$ to the origin $0=(0,0)$ is defined by

$$\tau_0 \stackrel{\text{def.}}{=} \inf\{n \ge 1 : Z(n) = (0,0)\}$$
and $(Z_{\tau_0}(n))$ denotes a process with the distribution of the Markov chain $(Z(n))$ killed at 0. Its Green function is defined by, for $j, k \in \mathbb{Z}_+^2$,

$$g(j, k) \stackrel{\text{def.}}{=} \sum_{n=0}^{\infty} \mathbb{P}_j(Z_{\tau_0}(n) = k) = \sum_{n=0}^{\infty} \mathbb{P}_j(Z(n) = k, \tau_0 > n). \quad (2.8)$$

A non-negative function $\varkappa: \mathbb{Z}_+^2 \to \mathbb{R}_+$ is said to be harmonic for the Markov chain $(Z(n))$ killed at 0 if, for $j \in \mathbb{Z}_+^2$,

$$\mathbb{E}_j(\varkappa(Z(1)); \tau_0 > 1) = \varkappa(j).$$

The next proposition introduces key quantities used to define the different regions which determine the asymptotic behavior of the Green function $(g(j, k))$. Its proof follows from Lemma 4.1 and Lemma 4.2 of Section 4. Figures 1, 2, 3 and 5 in Section C of the appendix illustrate some of these definitions.

**Proposition 2.1.** Under the assumptions (A1)-(A3),

(1) the sets $D_1$ and $D_2$ are logarithmically convex and the set $D$ is compact, strictly logarithmically convex and does not intersect with the axes $\{(x, y) \in \mathbb{R}^2: x=0\}$ and $\{(x, y) \in \mathbb{R}^2: y=0\}$;

(2) There exist $x_P^*, x_P^{**} \in ]0, +\infty[$ and $x^*, x^{**} \in ]0, +\infty[$ such that $x_P^* < x_P^{**}$, $x^* < x^{**}$, and

$$[x_P^*, x_P^{**}] = \{x \in ]0, +\infty[ : \exists y \in ]0, +\infty[ , (x, y) \in D\},$$

$$[x^*, x^{**}] = \{x \in ]0, +\infty[ : \exists y \in ]0, +\infty[ , (x, y) \in D \cap D_1\}.$$

(3) There exist functions $Y_1, Y_2: [x_P^*, x_P^{**}] \to [y_P^*, y_P^{**}]$, such that, for $x \in [x_P^*, x_P^{**}]$, $Y_1(x) \le Y_2(x)$ and

$$[Y_1(x), Y_2(x)] = \{y \in ]0, +\infty[ : (x, y) \in D\}.$$

For $x \in [x_P^*, x_P^{**}]$, $Y_1(x)$, $Y_2(x)$ are the unique positive solutions of the equation $P(x, y)=1$, and $Y_1(x)=Y_2(x)$ holds if and only if $x \in \{x_P^*, x_P^{**}\}$.

(4) There exist $y_P^*, y_P^{**} \in ]0, +\infty[$ and $y^*, y^{**} \in ]0, +\infty[$ such that $y_P^* < y_P^{**}$, $y^* < y^{**}$, and

$$[y_P^*, y_P^{**}] = \{y \in ]0, +\infty[ : \exists x \in ]0, +\infty[ , (x, y) \in D\},$$

$$[y^*, y^{**}] = \{x \in ]0, +\infty[ : \exists x \in ]0, +\infty[ , (x, y) \in D \cap D_2\}.$$

(5) There exist functions $X_1, X_2: [y_P^*, y_P^{**}] \to [x_P^*, x_P^{**}]$ such that, for $y \in [y_P^*, y_P^{**}]$, $X_1(y) \le X_2(y)$ and

$$[X_1(y), X_2(y)] = \{x \in ]0, +\infty[ : (x, y) \in D\}.$$

For $y \in [y_P^*, y_P^{**}]$, $X_1(y)$, $X_2(y)$ are the unique positive solutions of the equation $P(x, y)=1$, and $X_1(y)=X_2(y)$ holds if and only if $y \in \{y_P^*, y_P^{**}\}$.

The relations $D \cap D_1 \subset D$ and $D \cap D_2 \subset D$ give the inequalities

$$x_P^* \le x^* < x^{**} \le x_P^{**} \quad \text{and} \quad y_P^* \le y^* < y^{**} \le y_P^{**}. \qquad (2.9)$$

Note that, since the point $(1, 1)$ is an element of $D \cap D_1$ and $D \cap D_2$, one has also

$$x^* \le 1 \le x^{**}, \quad y^* \le 1 \le y^{**}. \qquad (2.10)$$
We now define four curves $S_{11}$, $S_{12}$, $S_{21}$ and $S_{22}$ on the boundary $\partial D$ of the set $D$. They are also used in the definition of our classification.

$$S_{11} \stackrel{\text{def.}}{=} \{(x, y) \in \partial D : \partial_x P(x, y) \le 0, \partial_y P(x, y) \le 0\}, \qquad (2.11)$$

$$S_{12} \stackrel{\text{def.}}{=} \{(x, y) \in \partial D : \partial_x P(x, y) \le 0, \partial_y P(x, y) \ge 0\}, \qquad (2.12)$$

$$S_{21} \stackrel{\text{def.}}{=} \{(x, y) \in \partial D : \partial_x P(x, y) \ge 0, \partial_y P(x, y) \le 0\}, \qquad (2.13)$$

$$S_{22} \stackrel{\text{def.}}{=} \{(x, y) \in \partial D : \partial_x P(x, y) \ge 0, \partial_y P(x, y) \ge 0\}. \qquad (2.14)$$

With Lemma 4.1 of Section 4, these curves can be expressed in terms of the functions $X_1$, $X_2$ and $Y_1$, $Y_2$ of Proposition 2.1 as follows,

$$S_{11} = \{(x, Y_1(x)) : x \in [x_P^*, X_1(y_P^*)]\} = \{(X_1(y), y) : y \in [y_P^*, Y_1(x_P^*)]\}, \qquad (2.15)$$

$$S_{12} = \{(x, Y_2(x)) : x \in [x_P^*, X_1(y_P^{**})]\} = \{(X_1(y), y) : y \in [Y_1(x_P^*), y_P^{**}]\}, \qquad (2.16)$$

$$S_{21} = \{(x, Y_1(x)) : x \in [X_1(y_P^*), x_P^{**}]\} = \{(X_2(y), y) : y \in [y_P^*, Y_1(x_P^{**})]\}, \qquad (2.17)$$

$$S_{22} = \{(x, Y_2(x)) : x \in [X_1(y_P^{**}), x_P^{**}]\} = \{(X_2(y), y) : y \in [Y_1(x_P^{**}), y_P^{**}]\}, \qquad (2.18)$$

and the relations $X_1(y_P^*)=X_2(y_P^*)$, $X_1(y_P^{**})=X_2(y_P^{**})$, $Y_1(x_P^*)=Y_2(x_P^*)$ and $Y_1(x_P^{**})=Y_2(x_P^{**})$ hold.

## 2.2 A Partition of the Space of Parameters

The next proposition shows that there is a partition of eight regions (Ba), $a \in \{0, \dots, 7\}$ for the possible locations of the points $x^{**}$, $Y_1(x^{**})$, $Y_2(x^{**})$, $y^{**}$, $X_1(y^{**})$, $X_2(y^{**})$. Its proof is given in Section 5. As it will be seen, the asymptotic behavior of the Green functions of the Markov chain killed at 0 depends on the region associated to its parameters. Figures 6, 7, 8 and 9 in Section C of the appendix give an illustration of several situations.

**Proposition 2.2 (Definition of the Classification).** Under the assumptions (A1)-(A3), one and only one of the following cases can occur,

$$(B0) \quad X_1(y^{**}) < x^{**} < X_2(y^{**}) \text{ and } Y_1(x^{**}) < y^{**} < Y_2(x^{**});$$

$$(B1) \quad X_1(y^{**}) < X_2(y^{**}) = x^{**} < x_P^{**}, \quad Y_1(x^{**}) < Y_2(x^{**}) = y^{**} < y_P^{**} \text{ and } (x^{**}, y^{**}) \in S_{22};$$

$$(B2) \quad X_2(y^{**}) < x^{**}, \quad Y_2(x^{**}) < y^{**} \text{ and } (x^{**}, Y_2(x^{**})), (X_2(y^{**}), y^{**}) \in S_{22};$$

$$(B3) \quad x^{**} = X_1(y^{**}) < x_P^{**}, \quad Y_1(x^{**}) < Y_2(x^{**}) = y^{**}, \quad y^{**} \le 1 < Y_2(x^{**}) \text{ and } (x^{**}, y^{**}) \in S_{12};$$

$$(B4) \quad x^{**} < X_1(y^{**}) < x_P^{**}, \quad Y_1(x^{**}) < Y_2(x^{**}) < y^{**}, \quad y^{**} \le 1 < Y_2(x^{**}) \text{ and} \\ (x^{**}, Y_2(x^{**})), (X_1(y^{**}), y^{**}) \in S_{12};$$

$$(B5) \quad y^{**} = Y_1(x^{**}) < y_P^{**}, \quad X_1(y^{**}) < X_2(y^{**}) = x^{**}, \quad x^{**} \le 1 < X_2(y^{**}) \text{ and } (x^{**}, y^{**}) \in S_{21};$$

$$(B6) \quad y^{**} < Y_1(x^{**}) < y_P^{**}, \quad X_1(y^{**}) < X_2(y^{**}) < x^{**}, \quad x^{**} \le 1 < X_2(y^{**}) \text{ and} \\ (X_2(y^{**}), y^{**}), (x^{**}, Y_1(x^{**})) \in S_{21};$$

$$(B7) \quad x^{**} = X_1(y^{**}) = 1, \quad y^{**} = Y_1(x^{**}) = 1, \quad \partial_x P(1, 1) < 0 \text{ and } \partial_y P(1, 1) < 0.$$

The cases (B0)-(B7) have in fact a simple geometrical interpretation. They are determined by the location of the line segments $[(x^{**}, Y_1(x^{**})), (x^{**}, Y_2(x^{**}))]$ and $[(X_1(y^{**}), y^{**}), (X_2(y^{**}), y^{**})]$. See Figures 5 of Section C of the appendix, where, see Section 5,
- the horizontal line segment represents $[(X_1(y^{**}), y^{**}), (X_2(y^{**}), y^{**})]$;

- the vertical segment $"[(x^{**}, Y_1(x^{**})), (x^{**}, Y_2(x^{**})]$.

The case (B7) corresponds to the case when the Markov chain $(Z(n))$ is transient and escapes to the infinity along the each of the axes $\{0\} \times \mathbb{N}$ and $\mathbb{N} \times \{0\}$. See Section A. In this case, the exact asymptotic of the Green function has been already obtained in the paper [19] by using methods of Markov-Additive processes. For this reason we will not consider this case.

Recall that in the literature, asymptotic results for nearest neighbor random walks in the quadrant were formulated either under conditions of positive recurrence, see [8, 24, 25, 26], or under conditions of transience, see [22], in terms of the recurrence properties of the random walks, i.e. via conditions satisfied by the drift vectors. It turns out, as we will see, that the drift vectors are not appropriate to define properly such a classification in the general case we consider. As it will be seen in Section A, the classification of Proposition 2.2 is not defined in such a way. The recurrence/transience properties of $(Z(n))$ have in fact a marginal impact in our investigation of the asymptotic behavior of the Green function $(g(j, k))$ of the killed Markov chain. Transience and recurrence properties may hold in each of the regions $(Ba)$, $a \in \{3, 4, 5, 6\}$. See Proposition A.1.

## 2.3 Convergence Domain and Functional Equation

For $j \in \mathbb{Z}_+^2$, the generating function of $(g(j, k), k \in \mathbb{Z}_+^2)$, defined on its convergence domain in $\mathbb{C}^2$, is denoted as

$$H_j(x, y) \stackrel{\text{def.}}{=} \sum_{k=(k_1, k_2) \in \mathbb{Z}_+^2 \setminus \{0\}} g(j, k) x^{k_1} y^{k_2}. \quad (2.19)$$

This is a central set of functions in our analysis. A significant part of our work is devoted to the investigation of their convergence domain and also to determine the nature of the dominant singularities of the functions $x \mapsto H_j(x, 0)$ and $y \mapsto H_j(0, y)$. Once it is done, with Tauberian like theorems and complex analysis arguments, we will able to derive the asymptotic behavior of the Green function $(g(j, k), k \in \mathbb{Z}_+^2)$ when $k$ goes to infinity.

Our first important result is that if

$$\Gamma \stackrel{\text{def.}}{=} \{(x, y) \in [0, +\infty]^2: x<x' \text{ and } y<y' \text{ for some } (x', y') \in D\}, \quad (2.20)$$

where $D$ is defined by (2.7), and

$$x_d \stackrel{\text{def.}}{=} \begin{cases} x^{**} & \text{if one of the conditions (B0)-(B4) holds,} \\ X_2(y^{**}) & \text{if either (B5) or (B6) holds,} \end{cases} \quad (2.21)$$

$$y_d \stackrel{\text{def.}}{=} \begin{cases} y^{**} & \text{if one of the conditions (B0)-(B2), (B5) or (B6) holds,} \\ Y_2(x^{**}) & \text{if either (B3) or (B4) holds.} \end{cases} \quad (2.22)$$

then, for any $j \in \mathbb{Z}_+^2$, the generating function $(x, y) \mapsto H_j(x, y)$ is analytic on

$$\Omega_d(\Gamma) \stackrel{\text{def.}}{=} \{(x, y) \in \Omega(\Gamma) : |x|<x_d, |y|<y_d\}, \quad (2.23)$$

In our next result, we will see that the point $x_d$ (resp. $y_d$) is the dominant singularity of the functions $x \mapsto H_j(x, 0)$ (resp. of the functions $y \mapsto H_j(0, y)$) and that the set $\Omega_d(\Gamma)$ is the maximal domain in $\mathbb{C}^2$ where all generating functions $(x, y) \mapsto H_j(x, y)$, $j \in \mathbb{Z}_+^2$, converge.

For $j \in \mathbb{Z}_+^2$, to formulate a key relation for the generating functions $(H_j(x, y))$, we define

$$Q(x, y) \stackrel{\text{def.}}{=} xy(1 - P(x, y)) = xy - \sum_{k=(k_1, k_2) \in \mathbb{Z}^2} x^{k_1+1} y^{k_2+1} \mu(k) \quad (2.24)$$
and

$$ \psi_1(x, y) \stackrel{\text{def.}}{=} x(1 - \phi_1(x, y)) = x - \sum_{k=(k_1, k_2) \in \mathbb{Z}^2} x^{k_1+1} y^{k_2} \mu_1(k), \quad (2.25) $$

$$ \psi_2(x, y) \stackrel{\text{def.}}{=} y(1 - \phi_2(x, y)) = y - \sum_{k=(k_1, k_2) \in \mathbb{Z}^2} x^{k_1} y^{k_2+1} \mu_2(k), \quad (2.26) $$

$$ L_j(x, y) \stackrel{\text{def.}}{=} \begin{cases} x^{j_1} y^{j_2} - \mathbb{P}_j(\tau_0 < +\infty) & \text{if } (j_1, j_2) \neq (0, 0), \\ \phi_0(x, y) - \mathbb{P}_{(0,0)}(\tau_0 < +\infty) & \text{if } (j_1, j_2) = (0, 0), \end{cases} \quad (2.27) $$

$$ h_j(x, y) \stackrel{\text{def.}}{=} \sum_{k=(k_1, k_2) \in \mathbb{Z}_+^2} g(j, (k_1+1, k_2+1)) x^{k_1} y^{k_2}, \quad (2.28) $$

$$ h_{1j}(x) \stackrel{\text{def.}}{=} \sum_{k_1=0}^{\infty} g(j, (k_1+1, 0)) x^{k_1}, \quad \text{and} \quad h_{2j}(y) \stackrel{\text{def.}}{=} \sum_{k_2=0}^{\infty} g(j, (0, k_2+1)) y^{k_2}. \quad (2.29) $$

Under the assumptions (A1)-(A3), the functions $(x, y) \mapsto L_j(x, y)$, $j \in \mathbb{Z}_+^2$ are clearly analytic in a neighborhood of the set $\Omega(\bar{\Gamma})$, see relation (2.2), and the functions $(x, y) \mapsto Q(x, y)$ and $(x, y) \mapsto \psi_i(x, y)$, $i=1, 2$, can be analytically continued to a neighborhood of $\Omega(\bar{\Gamma})$.

**Theorem 2.1 (Convergence Domain and Functional Equation).** Under the assumptions (A1)-(A4), for any $j \in \mathbb{Z}_+^2$, the following assertions hold

i) The function $x \mapsto h_{1j}(x)$, resp. $y \mapsto h_{2j}(y)$, is analytic in $B(0, x_d)$, resp. in $B(0, y_d)$.

ii) On the set $\Omega_d(\Gamma)$ the function $(x, y) \mapsto h_j(x, y)$ is analytic and the relation

$$ Q(x, y)h_j(x, y) = L_j(x, y) + \psi_1(x, y)h_{1j}(x) + \psi_2(x, y)h_{2j}(y) \quad (2.30) $$

holds.

When the random walk $(Z(n))$ is positive recurrent, Theorem 2.1 has been established in Kobayashi and Miyazawa [21]. With Proposition 2.2, the transience or recurrence properties do not play a role in our proof of this result.

Section 6 is devoted to the proof of Theorem 2.1. We give a sketch of it.

(1) We first prove that the series (2.29) and (2.28) converge in $\Omega(\Theta)$ for a logarithmically convex set $\Theta \subset [0, +\infty]^2$ whose boundary contains the line segments $[(x_d, 0), (x_d, Y_1(x_d))]$ and $[(0, y_d), (X_1(y_d), y_d)]$ and such that $\mathring{\Theta} \cap \mathring{D} \neq \emptyset$ and $\Theta \cup \mathring{D} = \{(x, y) \in \Gamma : x < x_d, y < y_d\}$. See Proposition 6.1 and Lemma 6.4. An important ingredient of the proof of this step is the use of Lyapunov functions. See also Figure 10 of Section C of the appendix.

(2) The functional equation (2.30) is established on the set $\Omega(\Theta)$. From there, we get that, for $j \in \mathbb{Z}_+^2$, the functions $(x, y) \mapsto Q(x, y)h_j(x, y)$ can be analytically continued to the set $\Omega_d(\Gamma)$.

(3) Since the function $(x, y) \mapsto h_j(x, y)$ is analytic in $\Omega(\Theta)$ and the function $(x, y) \mapsto 1/Q(x, y)$ is analytic in $\Omega(\mathring{D})$, from these results, we will be able to deduce that the function $(x, y) \mapsto h_j(x, y)$ can be continued as an analytic function to the set $\Omega_d(\Gamma)$.

**Remark.**

Clearly, for $j \in \mathbb{Z}_+^2$ and $x \neq 0, y \neq 0$,

$$ H_j(x, 0) = xh_{1j}(x), \quad H_j(0, y) = yh_{2j}(y), \quad H_j(x, y) - H_j(x, 0) - H_j(0, y) = xyh_j(x, y), $$
and the functional equation (1.2) of the introduction is equivalent to relation (2.30). The technical advantage of the formulation (2.30) is that the functions $\psi_1, \psi_2$ and $Q$ are also defined when $x=0$ or $y=0$. In some cases, however, the expression (1.2) is more convenient.

## 2.4 Singularity Analysis of $H_j$

Since the set $\Omega_d(\Gamma)$ can be represented as a union of the poly-discs centered at the origin of $\mathbb{C}^2$, the above theorem proves that for any $j \in \mathbb{Z}_+^2$, the generating function $H_j(x, y)$ converge in $\Omega_d(\Gamma)$. The results of this section establish that the set $\Omega_d(\Gamma)$ defined by (2.23) is the exact domain of convergence of the functions $H_j$, $j \in \mathbb{Z}_+^2$, i.e. the maximal domain in $\mathbb{C}^2$ where the series (2.19) converge, and the dominant singularities of the functions $x \mapsto H_j(x, 0)$ and $y \mapsto H_j(0, y)$ and their nature are identified.

Before stating these results, in order to have straight assertions, we have to fix a (small) technical problem related to irreducibility. It must be noted that, even under the assumptions (A1)-(A3), the killed Markov chain $(Z_{\tau_0}(n))$, see Definition 1, is not necessarily irreducible on the space $\mathbb{Z}_+^2 \setminus \{0\}$. One may have $g(j, k)=0$ for some $j, k \in \mathbb{Z}_+^2 \setminus \{0\}$. The following lemma solves this problem.

**Lemma 2.2.** Under the assumptions (A1)-(A3), there exist $N_0 \ge 0$ and a finite subset $E_0$ of $\mathbb{Z}_+^2 \setminus \{(0, 0)\}$ such that for $j, k \in \mathbb{Z}_+^2$,

$$g(j, k) = 0 \quad \text{if } j \in E_0 \text{ and } k \notin E_0, \qquad (2.31)$$

$$g(j, k) > 0 \quad \text{if } j \notin E_0 \text{ and } \|k\| \ge N_0. \qquad (2.32)$$

Consequently, to investigate the asymptotics of $g(j, k)$ as $k$ goes to infinity, it is sufficient to consider a starting point $j$ outside $E_0$. The proof of this lemma is given in Section B of the appendix. From now on, the integer $N_0 \ge 0$ and the set $E_0$ satisfying (2.31) and (2.32) are fixed. When the killed Markov chain $(Z_{\tau_0}(n))$ is irreducible on $\mathbb{Z}_+^2 \setminus \{(0, 0)\}$, the set $E_0$ is of course empty.

**Proposition 2.2** and the definition of the points $x_d$ and $y_d$ show that the relations, $x_d \le x^{**} \le x_P^{**}$ and $y_d \le y^{**} \le y_P^{**}$ hold. For $j \in \mathbb{Z}_+^2 \setminus E_0$, we will prove that $x_d$, resp. $y_d$, is the dominant singularity of the function $x \mapsto H_j(x, 0)$, resp. $y \mapsto H_j(0, y)$. We will see that the nature of the singularity $x_d$, resp. $y_d$, is determined by the cases (B0)-(B6) and also by several relations between the quantities $x_d, x^{**}$ and $x_P^{**}$, resp. $y_d, y^{**}$ and $y_P^{**}$.

By relation (2.9), the points $x^{**}$ and $y^{**}$ are respectively in $[x_P^{**}, x_P^{**}]$ and $[y_P^{**}, y_P^{**}]$, and from the definition of the functions $Y_2: [x_P^{**}, x_P^{**}] \to [y_P^{**}, y_P^{**}]$ and $X_2: [y_P^{**}, y_P^{**}] \to [x_P^{**}, x_P^{**}]$, we have

$$Y_2(x^{**}) \le y_P^{**} \quad \text{and} \quad X_2(y^{**}) \le x_P^{**}.$$

**Proposition 2.2** gives that $y^{**} < y_P^{**}$ if one of the cases (B0),(B1), (B5) or (B6) holds and, similarly, $x^{**} < x_P^{**}$ if one of the cases (B0)-(B4) holds. This is summarized as follows.

$$x_d = \begin{cases} x^{**} < x_P^{**} & \text{if one of the cases (B0),(B1), (B3) or (B4) holds,} \\ x^{**} \le x_P^{**} & \text{with a possible equality } x^{**} = x_P^{**}, \text{ if (B2) holds,} \\ X_2(y^{**}) = x^{**} \le x_P^{**} & \text{" } x^{**} = x_P^{**}, \text{ if (B5) holds,} \\ X_2(y^{**}) < x^{**} \le x_P^{**} & \text{if (B6) holds,} \end{cases} \qquad (2.33)$$

and

$$y_d = \begin{cases} y^{**} < y_P^{**} & \text{if one of the cases (B0),(B1), (B5) or (B6) holds,} \\ y^{**} \le y_P^{**} & \text{with a possible equality } y^{**} = y_P^{**}, \text{ if (B2) holds,} \\ Y_2(x^{**}) = y^{**} \le y_P^{**} & \text{" } y^{**} = y_P^{**}, \text{ if (B3) holds,} \\ Y_2(x^{**}) < y^{**} \le y_P^{**} & \text{if (B4) holds.} \end{cases} \qquad (2.34)$$
We now introduce several functions on $\mathbb{Z}_+^2$ which will be used to describe the dependence on the initial state $j \in \mathbb{Z}_+^2$ in the asymptotic behavior of $g(j, k)$ when $k$ goes to infinity. Under convenient conditions, as we will see, these functions are harmonic for the killed Markov chain at 0.

**Definition 2 (Functions for the Dependence on the Initial State).** For $j \in \mathbb{Z}_+^2$,

$$ \varkappa_1(j) \stackrel{\text{def.}}{=} L_j(x_d, Y_1(x_d)) + (\phi_2(x_d, Y_1(x_d)) - 1) H_j(0, Y_1(x_d)), \quad (2.35) $$

$$ \tilde{\varkappa}_1(j) \stackrel{\text{def.}}{=} \partial_y \left( \frac{L_j(x, y) + (\phi_2(x, y) - 1) H_j(0, y)}{1 - \phi_1(x, y)} \right) \Big|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))}, \quad (2.36) $$

$$ \varkappa_2(j) \stackrel{\text{def.}}{=} L_j(X_1(y_d), y_d) + (\phi_1(X_1(y_d), y_d) - 1) H_j(X_1(y_d), 0), \quad (2.37) $$

$$ \tilde{\varkappa}_2(j) \stackrel{\text{def.}}{=} \partial_x \left( \frac{L_j(x, y) + (\phi_1(x, y) - 1) H_j(x, 0)}{1 - \phi_2(x, y)} \right) \Big|_{(x,y)=(X_1(y_P^{**}), y_P^{**})}, \quad (2.38) $$

$$ \varkappa_{(x,y)}(j) \stackrel{\text{def.}}{=} L_j(x, y) + (\phi_1(x, y) - 1) H_j(x, 0) + (\phi_2(x, y) - 1) H_j(0, x), \quad (2.39) $$

By Theorem 2.1, and since the functions $\psi_1, \psi_2$, $Q$ and $L_j$ are analytic in a neighborhood of the set $\Omega(\bar{\Gamma})$, we have

- $\varkappa_1$ is well defined on $\mathbb{Z}_+^2$ if $Y_1(x_d)<x_d$, that is, if one of the cases (B0)-(B4) holds;

- $\tilde{\varkappa}_1$ if (B2), $x_d=x^{**}=x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**}))<1$ hold;

- $\varkappa_2$ if one of the cases (B0)-(B2), (B5) or (B6) holds;

- $\tilde{\varkappa}_2$ if (B2), $y_d=y^{**}=y_P^{**}$ and $\phi_2(X_1(y_P^{**}), y_P^{**})<1$ hold;

- $\varkappa_{(x,y)}$ for $(x, y) \in \Omega(\bar{\Gamma})$ such that $|x|<x_d$ and $|y|<y_d$.

The following theorem gives a complete description of the nature of the singularity $x_d$ for the function $x \mapsto H_j(x, 0)$. Note that the case (B7) is not considered because, as mentioned before, it has been already investigated in [19].

**Theorem 2.3 (Singularity Analysis of $(H_j)$).** Under the assumptions (A1)-(A3), the following assertions hold.

i) If one of the cases (B0), (B1), (B3), (B4) holds or (B2) and $x_d<x_P^{**}$ hold, then

- there exists $\varepsilon>0$ such that, for any $j \in \mathbb{Z}_+^2$, the function $x \mapsto H_j(x, 0)$ can be analytically continued to the set $B(0, x_d+\varepsilon) \setminus \{x_d\}$;

- the function $\varkappa_1$ of (2.35) is non-negative on $\mathbb{Z}_+^2$, harmonic for the Markov chain $(Z(n))$ killed at 0 and positive on the set $\mathbb{Z}_+^2 \setminus E_0$;

- for any $j \in \mathbb{Z}_+^2$,

$$ \lim_{x \to x_d} (x_d - x) H_j(x, 0) = a_1 \varkappa_1(j), \quad (2.40) $$

where

$$ a_1 = \left( \left. \frac{d}{dx} \phi_1(x, Y_1(x)) \right|_{x=x_d} \right)^{-1} > 0. \quad (2.41) $$

ii) If (B2) and $x_d=x_P^{**}$ hold, then

- there exists $\varepsilon>0$ such that, for any $j \in \mathbb{Z}_+^2$, the function $x \mapsto H_j(x, 0)$ can be analytically continued to $B(0, x_P^{**}+\varepsilon) \setminus [x_P^{**}, x_P^{**}+\varepsilon]$;
- if $\phi_1(x_d, Y_1(x_d))=1$, then the function $\varkappa_1$ is non-negative $\mathbb{Z}_+^2$, harmonic for the Markov chain $(Z(n))$ killed at 0, positive on the set $\mathbb{Z}_+^2 \setminus E_0$ and for any $j \in \mathbb{Z}_+^2$,

$$ \lim_{x \to x_d} \sqrt{x_d - x} H_j(x, 0) = a_2 \varkappa_1(j), \qquad (2.42) $$

where the limit is taken in the set $B(0, x_P^{**} + \varepsilon) \setminus [x_P^{**}, x_P^{**} + \varepsilon]$ and

$$ a_2 = \left( \partial_y \phi_1(x, y) \sqrt{\partial_x P(x, y) / \partial_{yy}^2 P(x, y)} \right)^{-1} \Bigg|_{(x,y)=(x_d, Y_1(x_d))} > 0; \qquad (2.43) $$

- if $\phi_1(x_d, Y_1(x_d))<1$, then the function $\tilde{\varkappa}_1$ of (2.36) is non-negative on $\mathbb{Z}_+^2$, harmonic for the Markov chain $(Z(n))$ killed at 0, positive on the set $\mathbb{Z}_+^2 \setminus E_0$, and for any $j \in \mathbb{Z}_+^2$,

$$ \lim_{x \to x_d} \sqrt{x_d - x} \frac{d}{dx} H_j(x, 0) = a_3 \tilde{\varkappa}_1(j), \qquad (2.44) $$

where the limit is taken in the set $B(0, x_P^{**} + \varepsilon) \setminus [x_P^{**}, x_P^{**} + \varepsilon]$ and

$$ a_3 = \frac{1}{2} \sqrt{\partial_x P(x, y) / \partial_{yy}^2 P(x, y)} \Bigg|_{(x,y)=(x_d, Y_1(x_d))} > 0. \qquad (2.45) $$

iii) If (B5) and $x_d<x_P^{**}$ hold, then

- there exists $\varepsilon>0$ such that, for any $j \in \mathbb{Z}_+^2$, the function $x \mapsto H_j(x, 0)$ can be analytically continued to the set $B(0, x_d + \varepsilon) \setminus \{x_d\}$;

- the function $\varkappa_2$ of (2.37) is non-negative on $\mathbb{Z}_+^2$, harmonic for the Markov chain $(Z(n))$ killed at 0, positive on the set $\mathbb{Z}_+^2 \setminus E_0$ and for any $j \in \mathbb{Z}_+^2$,

$$ \lim_{x \to x_d} (x_d - x)^2 H_j(x, 0) = a_4 \varkappa_2(j), \qquad (2.46) $$

where

$$ a_4 = (\phi_2(x, y) - 1) \left( \frac{d}{dx} \phi_1(x, Y_1(x)) \frac{d}{dy} \phi_2(X_1(y), y) \frac{d}{dx} Y_1(x) \right)^{-1} \Bigg|_{(x,y)=(x_d, y_d)} > 0. \qquad (2.47) $$

iv) If (B5) and $x_d=x_P^{**}$ hold, then

- there exists $\varepsilon>0$ such that, for any $j \in \mathbb{Z}_+^2$, the function $x \mapsto H_j(x, 0)$ can be continued as an analytic functions to the set $B(0, x_P^{**} + \varepsilon) \setminus [x_P^{**}, x_P^{**} + \varepsilon]$;

- the function $\varkappa_2$ is non-negative on $\mathbb{Z}_+^2$, harmonic for the Markov chain $(Z(n))$ killed at 0 and positive on the set $\mathbb{Z}_+^2 \setminus E_0$;

- if $\phi_1(x_d, Y_1(x_d))=1$ then for any $j \in \mathbb{Z}_+^2$,

$$ \lim_{x \to x_d} (x_d - x) H_j(x, 0) = a_5 \varkappa_2(j), \qquad (2.48) $$

where the limit is taken in the set $B(0, x_P^{**} + \varepsilon) \setminus [x_P^{**}, x_P^{**} + \varepsilon]$ and

$$ a_5 = (\phi_2(x, y) - 1) \partial_{yy}^2 P(x, y) \left( \partial_y \phi_1(x, y) \partial_x P(x, y) \frac{d\phi_2}{dy}(X_1(y), y) \right)^{-1} \Bigg|_{(x,y)=(x_d, y_d)} > 0. \qquad (2.49) $$
- if $\phi_1(x_d, Y_1(x_d))<1$ then for any $j \in \mathbb{Z}_+^2$,

$$ \lim_{x \to x_d} \sqrt{x_d - x} H_j(x, 0) = a_6 \varkappa_2(j), \qquad (2.50) $$

where the limit is taken in the set $B(0, x_P^{**}+\varepsilon) \setminus [x_P^{**}, x_P^{**}+\varepsilon]$ and

$$ a_6 = (\phi_2(x, y)-1) \sqrt{\frac{\partial_{yy}^2 P(x, y)}{\partial_x P(x, y)}} \left( (1-\phi_1(x, y)) \frac{d}{dy} \phi_2(X_1(y), y) \right)^{-1} \Bigg|_{(x,y)=(x_d, y_d)} > 0. \qquad (2.51) $$

v) If (B6) holds, then

- there exists $\varepsilon>0$ such that, for any $j \in \mathbb{Z}_+^2$, the function $x \mapsto H_j(x, 0)$ can be analytically continued to the set $B(0, x_d+\varepsilon) \setminus \{x_d\}$;

- the function $\varkappa_2$ is non-negative on $\mathbb{Z}_+^2$, harmonic for the Markov chain $(Z(n))$ killed at 0, positive on the set $\mathbb{Z}_+^2 \setminus E_0$ and for any $j \in \mathbb{Z}_+^2$,

$$ \lim_{x \to x_d} (x_d - x) H_j(x, 0) = a_7 \varkappa_2(j), \qquad (2.52) $$

where

$$ a_7 = (\phi_2(x, y)-1) \left( (1-\phi_1(x, y)) \frac{d}{dy} \phi_2(X_1(y), y) \frac{d}{dx} Y_1(x) \right)^{-1} \Bigg|_{(x,y)=(x_d, y_d)} > 0. \qquad (2.53) $$

vi) If (B2) holds, then

- the set $\{(x, y) \in S_{22} : x<x_d, y<y_d\}$ is non-empty;

- there exists a neighborhood $\mathcal{V}$ of the set $S_{22}$ in $\mathbb{R}_+^2$ such that, for any $j \in \mathbb{Z}_+^2$, the function $x \mapsto (1-P(x, y))H_j(x, y)$ can be analytically continued to the set $\{(x, y) \in \mathcal{V} : |x|<x_d, |y|<y_d\}$;

- for any $(\hat{x}, \hat{y}) \in \{(x, y) \in S_{22} : x<x_d, y<y_d\}$, the function $\varkappa_{(\hat{x}, \hat{y})}$ is non-negative on $\mathbb{Z}_+^2$, harmonic for the Markov chain $(Z(n))$ killed at 0, positive on the set $\mathbb{Z}_+^2 \setminus E_0$ and for any $j \in \mathbb{Z}_+^2$,

$$ \lim_{\substack{(x,y) \to (\hat{x}, \hat{y}) \\ (x,y) \in \tilde{D}}} (1-P(x,y)) H_j(x,y) = \varkappa_{(\hat{x}, \hat{y})}(j). \qquad (2.54) $$

By symmetry (it is sufficient to to exchange the roles of $x$ and $y$), the analogous results for the assertions i)-v) of Theorem 2.3 hold for the functions $y \mapsto H_j(0, y)$, $j \in \mathbb{Z}_+^2 \setminus E_0$.

The last assertion of Theorem 2.3 is obtained as a consequence of Theorem 2.1. The proofs of the first five assertions are more demanding as it will be seen.

### Remark.

In the context of the functional equation (2.30), a classical approach of the literature consists in finding a suitable analytic function $x \mapsto Y(x)$ (resp. $y \mapsto X(y)$) satisfying the equation $Q(x, Y(x))=0$ for any $x$, resp. $Q(X(y), y)=0$ for any $y$, in some domain large enough, in order to inject $y=Y(x)$ (resp $x=X(y)$) in (2.30).

For nearest neighbor random walks, see Malyshev [25], the equation $Q(x, y)=0$ is quadratic in $x$ and $y$, its solutions have therefore an explicit form. In the general case, when the jumps of the random walk are unbounded, one clearly cannot find the functions $Y$ and $X$ in such a way. See Remark 2.3 of Kobayashi and Miyazawa [21].
## Probabilistic Representations of $X_1$ and $Y_1$

In our analysis, a part of the technicalities of the literature related to analytic continuation of the functions $(Y(x))$ and $(X(y))$ mentioned above is avoided via a probabilistic argument.

We show (see Section 7.2) that the functions $Y_1:[x_P^*, x_P^{**}] \to [y_P^*, y_P^{**}]$ and $X_1:[y_P^*, y_P^{**}] \to [x_P^*, x_P^{**}]$, introduced in Proposition 2.1, have a probabilistic representation. With this result, we are able to prove that there exists $\varepsilon>0$ such that the function $X_1$, resp $Y_1$, can be analytically continued to the set

$$ \{x \in \mathbb{C}: x_P^* < |x| < x_P^{**} + \varepsilon, x \notin [x_P^{**}, x_P^{**} + \varepsilon]\}, \text{ resp. } \{y \in \mathbb{C}: y_P^* < |y| < y_P^{**} + \varepsilon, y \notin [y_P^{**}, y_P^{**} + \varepsilon]\}. $$

From there, several important properties of the analytic continuation of $Y_1$ and $X_1$ are then derived. As a consequence one can inject $y=Y_1(x)$ or $x=X_1(y)$ in the equation (2.30), and then to establish Theorem 2.3.

## 2.5 Asymptotics of the Green function along the Axes

In this section, we investigate the asymptotics of the Green function $g(j, k)$ when one of coordinate of $k=(k_1, k_2) \in \mathbb{Z}_+^2$ is fixed, i.e. as $k_1 \to +\infty$ with $k_2$ fixed or $k_2 \to +\infty$ and $k_1$ fixed. By symmetry it is enough to consider only the first convergence.

We define $\nu_1(0)=1$ and, for $n \ge 1$,

$$ \nu_1(n) \stackrel{\text{def.}}{=} \frac{1}{(n-1)!} \left. \frac{\partial^{n-1}}{\partial y^{n-1}} \left( \frac{\psi_1(x_d, y)}{Q(x_d, y)} \right) \right|_{y=0}, \quad (2.55) $$

where $Q$ and $\psi_1$ are defined by (2.24) and (2.25) respectively. As the following theorem shows, the quantity $\nu_1(k_2)$ expresses the dependence on $k_2$ in the limiting behavior of $k_1 \to g(j, (k_1, k_2))$ when $k_1 \to +\infty$.

In Section 8, it is shown that $\nu_1$ is, up to a multiplicative constant, the invariant distribution of a twisted version of a random walk on $\mathbb{Z} \times \mathbb{Z}_+$ obtained by removing the boundary $\{0\} \times \mathbb{Z}_+$. It will show in particular that the coefficients $\nu_1(n)$, $n \in \mathbb{Z}_+$, are positive.

The case (B7), already analyzed, excepted, the following result gives a complete description of all possible cases for the asymptotic behavior of the Green function $g(j, (k_1, k_2))$ as $k_1 \to +\infty$ for a fixed $k_2 \in \mathbb{Z}_+$.

**Theorem 2.4 (Asymptotics of Green Function with a Fixed Second Component).** Under the assumptions (A1)-(A3), the following assertions hold.

(1) If either one of the cases (B0), (B1), (B3), (B4) holds or (B2) and $x_d < x_P^{**}$ hold, then for any $j \in \mathbb{Z}_+^2 \setminus E_0$, as $k_1 \to +\infty$ and $k_2$ varies in a finite subset of $\mathbb{Z}_+$,

$$ g(j, (k_1, k_2)) \sim a_1 \nu_1(k_2) \chi_1(j) x_d^{-k_1-1}. \quad (2.56) $$

where $\chi_1(j) > 0$ and $a_1 > 0$ are defined respectively by (2.35) and (2.41).

(2) If (B2) and $x_d = x_P^{**}$ hold, then for any $j \in \mathbb{Z}_+^2 \setminus E_0$, as $k_1 \to +\infty$ and $k_2$ varies in a finite subset of $\mathbb{Z}_+$

$$ g(j, (k_1, k_2)) \sim a_2 \nu_1(k_2) \chi_1(j) x_d^{-k_1} \left( \sqrt{\pi k_1 x_d} \right)^{-1} \quad \text{if } \phi_1(x_d, Y_1(x_d))=1, \quad (2.57) $$

$$ g(j, (k_1, k_2)) \sim a_3 \nu_1(k_2) \tilde{\chi}_1(j) x_d^{-k_1} \left( k_1 \sqrt{\pi k_1 x_d} \right)^{-1} \quad \text{if } \phi_1(x_d, Y_1(x_d))<1, \quad (2.58) $$

where $\tilde{\chi}_1(j) > 0$, $\chi_1(j) > 0$, $a_1 > 0$ and $a_2 > 0$ are defined respectively by (2.35), (2.36), (2.43) and (2.45).
(3) If (B5) holds and $x_d<x_P^{**}$, then for $j \in \mathbb{Z}_+^2 \setminus E_0$, as $k_1 \to +\infty$ and $k_2$ varies in a finite subset of $\mathbb{Z}_+$,

$$g(j, (k_1, k_2)) \sim a_4 \nu_1(k_2) \varkappa_2(j) k_1 x_d^{-k_1-2}, \qquad (2.59)$$

where $\varkappa_2(j) > 0$ and $a_4 > 0$ are defined by (2.37) and (2.47).

(4) If (B5) holds and $x_d=x_P^{**}$, then for $j \in \mathbb{Z}_+^2 \setminus E_0$, as $k_1 \to +\infty$ and $k_2$ varies in a finite subset of $\mathbb{Z}_+$,

$$g(j, (k_1, k_2)) \sim a_5 \nu_1(k_2) \varkappa_2(j) x_d^{-k_1-1} \quad \text{if } \phi_1(x_d, Y_1(x_d))=1, \qquad (2.60)$$

$$g(j, (k_1, k_2)) \sim a_6 \nu_1(k_2) \varkappa_2(j) x_d^{-k_1} \left(\sqrt{\pi k_1 x_d}\right)^{-1} \quad \text{if } \phi_1(x_d, Y_1(x_d))<1, \qquad (2.61)$$

where $\varkappa_2(j) > 0$, $a_5 > 0$ and $a_6 > 0$ are defined respectively by (2.37), (2.49) and (2.51).

(5) If (B6) holds, then for $j \in \mathbb{Z}_+^2 \setminus E_0$, as $k_1 \to +\infty$ and $k_2$ varies in a finite subset of $\mathbb{Z}_+$,

$$g(j, (k_1, k_2)) \sim a_7 \nu_1(k_2) \varkappa_2(j) x_d^{-k_1-1}, \qquad (2.62)$$

where $\varkappa_2(j) > 0$ and $a_7 > 0$ are defined respectively by (2.37) and (2.53).

For $k_2 = 0$, this result is obtained as a straightforward consequence of Theorem 2.3 by using the Tauberian-like theorem due to Flajolet and Odlyzko [9], see Corollary VI.1 of [10]). To get this result for $k_2 \in \mathbb{Z}_+$ that varies in a finite subset of $\mathbb{Z}_+$, it is sufficient to prove it for any given $k_2 \in \mathbb{Z}_+$. For this we prove that for any $j \in \mathbb{Z}_+^2 \setminus E_0$ and $k_2 \in \mathbb{Z}_+$,

$$\lim_{n \to +\infty} g(j, (n, k_2))/g(0, (n, k_2)) = \nu_1(k_2), \quad k_2 \in \mathbb{Z}_+,$$

by using a probabilistic representation of the coefficients $\nu_1(k_2)$, $k_2 \in \mathbb{Z}_+$. The proof of this result is given in Section 8.

## 2.6 Asymptotics of the Green function along Directions of $\mathbb{S}_+$

In this section, we present the asymptotics of the Green function $g(j, (k_1, k_2))$ as $\min\{k_1, k_2\} \to \infty$ and $k/\|k\| \to w$ where $w$, the direction, is an element of $\mathbb{S}_+^1 = \{(u_1, u_2) \in \mathbb{R}_+^2 : u_1^2+u_2^2=1\}$.

For each of the cases (B0)-(B6), we will introduce subsets $W_0$, $W_1$ and $W_2$ of $\mathbb{S}_+$ used to define the partition of the set of directions in $\mathbb{S}_+^1$. This is achieved by the definitions 3, 4, 5 and 6 in Section 2.6.2. A critical direction $w_c=(w_c, v_c)$ in $\mathbb{S}_+$ will play a role in several cases. As it will be seen this partition will determine a structure of the asymptotics behavior of the Green function $k \mapsto g(j, k)$.

### 2.6.1 Diffeomorphism between $\mathbb{S}_+$ and $S_{22}$

The sets $\mathbb{S}_+$ and $S_{22}$ are defined by (2.1) and (2.14). Under Assumptions (A1), see for instance Ney and Spitzer [27], the Laplace transform

$$(\alpha, \beta) \mapsto \tilde{P}(\alpha, \beta) \stackrel{\text{def.}}{=} P(e^{\alpha}, e^{\beta}) = \sum_{k=(k_1, k_2) \in \mathbb{Z}^2} \mu(k) e^{\alpha k_1 + \beta k_2}. \qquad (2.63)$$

is strictly convex, the level set $\tilde{D} \stackrel{\text{def.}}{=} \{(\alpha, \beta) \in \mathbb{R}^2 : \tilde{P}(\alpha, \beta) \le 1\}$ is strictly convex and compact, the gradient $\nabla_{\alpha, \beta} \tilde{P}(\alpha, \beta)$ does not vanishes on the boundary $\partial \tilde{D}$ of $\tilde{D}$, and the function

$$(\alpha, \beta) \mapsto \nabla_{\alpha, \beta} \tilde{P}(\alpha, \beta) / \|\nabla_{\alpha, \beta} \tilde{P}(\alpha, \beta)\| \qquad (2.64)$$
determines a diffeomorphism from $\partial \tilde{D}$ to the unit circle $\mathbb{S}^1=\{(u_1, u_2) \in \mathbb{R}^2 : u_1^2+u_2^2=1\}$.
Since for $x=e^\alpha$ and $y=e^\beta$, one has

$$ \partial_{\alpha} \tilde{P}(\alpha, \beta) = x \partial_x P(x, y) \quad \text{and} \quad \partial_{\beta} \tilde{P}(\alpha, \beta) = y \partial_y P(x, y), \qquad (2.65) $$

for the set $D$ defined by (2.7), it follows that the function

$$ (x, y) \mapsto w_D(x, y) = (u_D(x, y), v_D(x, y)) \\ \stackrel{\text{def.}}{=} \frac{1}{\sqrt{x^2(\partial_x P(x, y))^2 + y^2(\partial_y P(x, y))^2}} (x \partial_x P(x, y), y \partial_y P(x, y)) \quad (2.66) $$

is a diffeomorphism from $\partial D$ to $\mathbb{S}^1$ and, by the definition of $S_{22}$, from $S_{22}$ to $\mathbb{S}_+^1$. We denote by $w \mapsto (x_D(w), y_D(w))$, resp. $w \mapsto (\alpha_D(w), \beta_D(w))$, the inverse mapping of the function (2.66), resp. of (2.64).

Remark that, for $w_1=(u_1, v_1)$ and $w_2=(u_2, v_2) \in \mathbb{S}_+^1$, the inequalities $u_1<u_2$ and $v_1>v_2$ are equivalent and, since the set $\tilde{D}$ is strictly convex, $\alpha_D(u_1, v_1)<\alpha_D(u_2, v_2)$, resp. $\beta_D(u_1, v_1)<\beta_D(u_2, v_2)$, if and only if $u_1<u_2$, resp. $v_1>v_2$. Hence, using again (2.65) for $x=e^\alpha$ and $y=e^\beta$, one gets a similar property for the diffeomorphism $w \mapsto (x_D(w), y_D(w))$:

**Lemma 2.5.** Under Assumption (A1), for any $w_1=(u_1, v_1)$, $w_2=(u_2, v_2) \in \mathbb{S}_+^1$, then

$$ x_D(u_1, v_1) < x_D(u_2, v_2) \Leftrightarrow u_1 < u_2 \Leftrightarrow v_1 > v_2 \Leftrightarrow y_D(u_1, v_1) > y_D(u_2, v_2). $$

This elementary property of the diffeomorphism $w \mapsto (x_D(w), y_D(w))$ will be useful in the next section.

### 2.6.2 Regions of directions

**Definition 3.** If (B0) holds, we introduce the vector $w_c=(u_c, v_c) \in \mathbb{S}_+^1$ orthogonal to the vector

$$ (\ln x^{**} - \ln X_2(y^{**}), \ln X_2(y^{**}) - \ln y^{**}) $$

and we let

$$ W_1 = \{w=(u, v) \in \mathbb{S}_+^1 : u > u_c\} \qquad (2.67) $$

$$ W_2 = \{w=(u, v) \in \mathbb{S}_+^1 : u < u_c\} \qquad (2.68) $$

$$ W_0 = \emptyset. \qquad (2.69) $$

Since in the case (B0), we have $X_2(y_d)>x_d$ and $Y_2(x_d)>y_d$, such a vector $w_c=(u_c, v_c) \in \mathbb{S}_+^1$ exists, is clearly unique and has positive coordinates. The sets $W_1$ and $W_2$ are therefore both non-empty: $W_1$ is a part of $\mathbb{S}_+^1$ included in the half-plane

$$ \{w=(u, v) \in \mathbb{R}^2 : v_c u > u_c v\} = \{w=(u, v) \in \mathbb{R}^2 : (\ln(X_2(y_d)) - \ln(x_d))u > (\ln(Y_2(x_d)) - \ln(y_d))v\} $$

and contains the vector $(1, 0)$, and $W_2$ is a part of $\mathbb{S}_+^1$ included in the half-plane

$$ \{w=(u, v) \in \mathbb{R}^2 : v_c u < u_c v\} = \{w=(u, v) \in \mathbb{R}^2 : (\ln(X_2(y_d)) - \ln(x_d))u < (\ln(Y_2(x_d)) - \ln(y_d))v\} $$

and contains the vector $(0, 1)$.

**Definition 4.** If(B1) holds, we take $w_c=(u_c, v_c)=w_D(x_d, Y_2(x_d))$ and define the regions $W_0$, $W_1$ and $W_2$ in the same way as in the case (B0).
In the case (B1), we have $x_d<x_P^{**}$, $y_d<y_P^{**}$ and $(x_d, Y_2(x_d))=(X_2(x_d), y_d)\in S_{22}$, hence, similarly to the case (B0), both coordinates of the vector $w_c$ are positive and the sets of the regions $W_1$ and $W_2$ are non-empty: $W_1$ is a part of $S_+^1$ included in the half plane $\{w=(u, v) \in \mathbb{R}^2 : u_c u>v_c v\}$ containing the vector $(1, 0)$, and $W_2$ is a part of $S_+^1$ included in the half-plane $\{w=(u, v) \in \mathbb{R}^2 : u_c u<v_c v\}$ containing the vector $(0, 1)$.

**Definition 5.** If (B2) holds, we define

$$W_0 = \{w=(u, v) \in S_+^1 : u_D(X_2(y_d), y_d) < u < u_D(x_d, Y_2(x_d))\}, \quad (2.70)$$

$$W_1 = \{w=(u, v) \in S_+^1 : u > u_D(x_d, Y_2(x_d))\}, \quad (2.71)$$

$$W_2 = \{w=(u, v) \in S_+^1 : u < u_D(X_2(y_d), y_d)\}. \quad (2.72)$$

By Lemma 2.5, for $w=(u, v)\in S_+^1$, the inequalities $u_D(X_2(y_d), y_d)<u<u_D(x_d, Y_2(x_d))$ are equivalent to the inequalities $x_D(u, v)<x_d$ and $y_D(u, v)<y_d$, and hence, the above definition of the sets $W_0$, $W_1$ and $W_2$ is equivalent to

$$W_0 = \{w_D(x, y) : (x, y) \in S_{22}, x<x_d, y<y_d\},$$

$$W_1 = \{w_D(x, y) : (x, y) \in S_{22}, x>x_d\},$$

$$W_0 = \{w_D(x, y) : (x, y) \in S_{22}, y>y_d\}.$$

In the case (B2), we have $X_2(y_d)<x_d\le x_P^{**}$ and $Y_2(x_d)<y_d\le y_P^{**}$, hence the set $\{(x, y)\in S_{22} : x<x_d, y<y_d\}$ is non-empty, and consequently, the set of directions $W_0$ is also non-empty: this is a part of the set $S_+^1$ included in the intersection of the two half-planes $\{w = (u, v)\in \mathbb{R}^2 : v_D(X_2(y_d), y_d)u > u_D(X_2(y_d), y_d)v\}$ and $\{w=(u, v) \in \mathbb{R}^2 : v_D(x_d, Y_2(x_d))u < u_D(x_d, Y_2(x_d))v\}$. Similar arguments show that in this case,

- the set of directions $W_1$ is empty if and only if $x_d=x_P^{**}$,

- if and only if $y_d=y_P^{**}$.

**Definition 6.** If either (B3) or (B4) holds we let $W_2 = W_0 = \emptyset$ and we define $W_1$ by

$$W_1 = \begin{cases} S_+^1 \setminus \{(0, 1)\}, & \text{if (B3) holds} \\ S_+^1, & \text{if (B4) holds.} \end{cases} \quad (2.73)$$

Similarly, if either (B5) or (B6) holds we take $W_1=W_0=\emptyset$ and we define $W_2$ by

$$W_2 = \begin{cases} S_+^1 \setminus \{(1, 0)\}, & \text{if (B5) holds} \\ S_+^1, & \text{if (B6) holds.} \end{cases} \quad (2.74)$$

We will see that for any $j\in \mathbb{Z}_+^2 \setminus E_0$, the asymptotic of the Green function $g(j, k)$ as $\min\{k_1, k_2\}\to+\infty$ with $(k_1, k_2)/\|(k_1, k_2)\|\to w\in S_+^1$, is

- determined by the simple pole $x_d$ of the function $x\mapsto H_j(x, 0)$ when $w \in W_1$;

- $y_d$ when $w \in W_2$;

- similar to the asymptotic of the Green function of the homogeneous random walk on $\mathbb{Z}^2$ associated to the distribution $\mu$ when $w\in W_0$.
## Twisted Homogeneous Random Walks

To formulate convergence results when $\min\{k_1, k_2\} \to +\infty$ and $(k_1, k_2)/\|(k_1, k_2)\| \to w$, for $w \in W_0$, some quantities, also used in the asymptotics of the Green function of the homogeneous random walk, are now introduced. See Ney and Spitzer [27] or Theorem 25.15 in Woess [30].

**Definition 7.** For $w \in \mathbb{S}_+^1$, we denote by $(S^w(n))$ the homogeneous random walk on $\mathbb{Z}^2$ with transition probabilities, for $m, k=(k_1, k_2) \in \mathbb{Z}^2$,

$$ \mathbb{P}_m(S^w(1)=m+k) = (x_D(w))^{k_1}(y_D(w))^{k_2}\mu(k). $$

The vector of first moments and the matrix of second moments of $(S^w(n))$ are denoted respectively by $m(w)=(m_1(w), m_2(w))$ and $Q(w)=(Q_{i,j}(w))_{i,j=1}^2$: for $i, j \in \{1, 2\}$,

$$ m_i(w) = \sum_{k \in \mathbb{Z}^2} k_i (x_D(w))^{k_1} (y_D(w))^{k_2} \mu(k), $$

$$ Q_{i,j}(w) = \sum_{k \in \mathbb{Z}^2} k_i k_j (x_D(w))^{k_1} (y_D(w))^{k_2} \mu(k). $$

The associated quadratic form at $z \in \mathbb{R}^2$ is denoted by $z \cdot Q(w) \cdot z$. For $w=(u, v) \in \mathbb{S}^1$, we define $w^\perp=(-v, u)$.

We can now state our second set of asymptotic results. The proof of this theorem is given in Section 9. Recall that here and throughout the paper, for $k=(k_1, k_2) \in \mathbb{Z}^2$, we denote $w_k=k/\|k\|$. Theorem 2.6 (Asymptotics of Green Function along Directions of $\mathbb{S}_+^1$). Under the assumptions (A1)-(A3), for any $j \in \mathbb{Z}_+^2 \setminus E_0$, the following assertions hold:

i) If $\min\{k_1, k_2\} \to +\infty$, then, uniformly with respect to $w_k$ in any compact subset of $W_1$,

$$ g(j, k) \sim b_1 \chi_1(j) x_d^{-k_1-1} (Y_2(x_d))^{-k_2-1}, \quad (2.75) $$

where

$$ b_1 = (\phi_1(x_d, Y_2(x_d))-1) \left( \left. \partial_y P(x, y) \right|_{(x,y)=(x_d, Y_2(x_d))} \frac{d}{dx} \phi_1(x, Y_1(x)) \right|_{x=x_d} \right)^{-1} > 0; \quad (2.76) $$

ii) If $\min\{k_1, k_2\} \to +\infty$, then, uniformly with respect to $w_k$ in any compact subset of $W_2$,

$$ g(j, k) \sim b_2 \chi_2(j) (X_2(y_d))^{-k_1-1} y_d^{-k_2-1}, \quad (2.77) $$

where

$$ b_2 = (\phi_2(X_2(y_d), y_d)-1) \left( \left. \partial_x P(x, y) \right|_{(x,y)=(X_2(y_d), y_d)} \frac{d}{dy} \phi_2(X_1(y), y) \right|_{y=y_d} \right)^{-1} > 0. \quad (2.78) $$

iii) If (B0) holds then, as $\min\{k_1, k_2\} \to +\infty$ and $w_k \to w_c$,

$$ g(j, k) \sim C_1 \chi_1(j) x_d^{-k_1-1} (Y_2(x_d))^{-k_2-1} + C_2 \chi_2(j) (X_2(y_d))^{-k_1-1} y_d^{-k_2-1}, \quad (2.79) $$

where $b_1>0$ and $b_2>0$ are given respectively by (2.76) and (2.78).

iv) If (B3) and $y_d<y_P^{**}$ hold, then as $\min\{k_1, k_2\} \to +\infty$ and $w_k \to (0, 1)$,

$$ g(j, k) \sim \chi_1(j) \left( b_1 x_d^{-k_1-1} + b_3 (X_2(y_d))^{-k_1-1} y_d^{-k_2-1} k_2 \right) y_d^{-k_2-1}, \quad (2.80) $$
where $b_1 > 0$ is given by (2.76) and

$$b_3 = x_d(\phi_1(x_d, y_d)-1) \left. \left( \partial_y P(x, y) \frac{d}{dy} \phi_2(X_1(y), y) \frac{d}{dx} \phi_1(x, Y_1(x)) \frac{d}{dy} X_1(y) \right)^{-1} \right|_{(x,y)=(x_d, y_d)} > 0. \quad (2.81)$$

v) If (B5) and $x_d < x_P^{**}$ hold, then, as $\min\{k_1, k_2\} \to +\infty$ and $w_k \to (1, 0)$,

$$g(j, k) \sim \kappa_2(j) x_d^{-k_1-1} \left( b_4 k_1 x_d^{-1} (Y_2(x_d))^{-k_2-2} + b_2 y_d^{-k_2-1} \right) \quad (2.82)$$

where $b_2 > 0$ is given by (2.78) and

$$b_4 = y_d(\phi_2(x_d, y_d)-1) \left. \left( \partial_x P(x, y) \frac{d}{dx} \phi_1(x, Y_1(x)) \frac{d}{dy} \phi_2(X_1(y), y) \frac{d}{dx} Y_1(x) \right)^{-1} \right|_{(x,y)=(x_d, y_d)} > 0. \quad (2.83)$$

vi) if (B2) holds and $\min\{k_1, k_2\} \to +\infty$, then, uniformly with respect to $w_k$ in any compact subset of $W_0$,

$$g(j, k) \sim \kappa_{(x_D(w_k), y_D(w_k))}(j) \frac{\|\mathbf{m}(w_k)\| \sqrt{w_k^{\perp} \cdot Q(w_k) \cdot w_k^{\perp}}}{\sqrt{2\pi \|k\|} (x_D(w_k))^{k_1} (y_D(w_k))^{k_2}}. \quad (2.84)$$

### 2.6.3 Missing Asymptotics

With the definition of the regions of directions $W_0$, $W_1$ and $W_2$, the above theorem provides the asymptotics of the Green function $g(j, k)$ as $\min\{k_1, k_2\} \to +\infty$ for all possible directions $w$ except the following singular cases:

- (B1) holds and $w=w_c=w_D(x^{**}, Y_2(x^{**}))$;

- (B2) holds and either $w=w_D(x^{**}, Y_2(x^{**}))$ or $w=w_D(X_2(y^{**}), y^{**})$;

- (B3) and $y_d=y^{**}=y_P^{**}$ hold and $w=(0, 1)$;

- (B5) and $x_d=x^{**}=x_P^{**}$ hold and $w=(1, 0)$.

We believe that obtaining these missing asymptotics requires an additional significant, non-trivial, technical effort. It should be noted that for random walks, the asymptotics for these singular directions have not been derived, even in the case of nearest neighbor jumps. In Kurkova and Malyshev [22] such asymptotic results are obtained but only along lines of $Z_+^2$ with a rational direction. For reflected Brownian motions in $\mathbb{R}_+^2$, Franceschi et al. in their recent paper [13] proves such a result in the case analogous to (B2) for the directions $w=w_D(x^{**}, Y_2(x^{**}))$ and $w=w_D(X_2(y^{**}), y^{**})$. Their results show that the asymptotic behavior for these directions is expressed as the competition of two terms: one determined by a simple pole and another given by a saddle point, the slowest of them determines the exact asymptotics. We believe that in our setting a similar phenomenon should also hold. However, contrary to the case of reflected Brownian motions in $\mathbb{R}_+^2$, the roots of the kernel equation $Q(x, y)=0$ are not explicitly known in our case and hence, proving an analogous result seems to be quite challenging.

### Remarks on Asymptotic Expressions.

To the best of our knowledge, the asymptotic behaviors iv) and v) have not been established in the literature, even in the case of nearest neighbor jumps. Note that the asymptotics of the cases iii), iv) and v) of Theorem 2.6 are expressed as a sum of two terms. We now discuss the implications of these asymptotic results.
Case iii) The asymptotics (2.79) reflect in fact a competition between the geometric decay determined by the simple pole $x_d=x^{**}$ of the function $x\mapsto H_j(x, 0)$ and the geometric decay determined by the simple pole $y_d=y^{**}$ of the function $y\mapsto H_j(0, y)$. In the case when (B0) holds and if $(k_n)=(k_{1,n}, k_{2,n})$ is a sequence of points of $\mathbb{Z}_+^2$ going to infinity such that $k_n/\|k_n\|\to w_c$, since $(x_d/X_2(y_d))^{u_c}=(y_d/Y_2(x_d))^{v_c}$ and $X_2(y_d)>x_d$, Theorem 2.6 shows in fact that for $j\in\mathbb{Z}_+^2\setminus E_0$,

$$
\begin{align}
g(j, k_n) &\sim C_1 \varkappa_1(j) x_d^{-k_{1,n}-1} (Y_2(x_d))^{-k_{2,n}-1} \quad \text{if } \lim_{n} (k_{1,n}-k_{2,n} u_c / v_c) = +\infty \nonumber \\
g(j, k_n) &\sim C_2 \varkappa_2(j) (X_2(y_d))^{-k_{1,n}-1} y_d^{-k_{2,n}-1} \quad \text{if } \lim_{n} (k_{2,n}-k_{2,n} u_c / v_c) = -\infty \nonumber \\
g(j, k_n) &\sim \left( C_1 \varkappa_1(j) + C_2 \left( \frac{x_d}{X_2(y_d)} \right)^\sigma \varkappa_2(j) \right) x_d^{-k_{1,n}-1} (Y_2(x_d))^{-k_{2,n}-1} \tag{2.85} \\
&\qquad \text{if } \lim_{n} (k_{1,n}-k_{2,n} u_c / v_c) = \sigma \in \mathbb{R}. \nonumber
\end{align}
$$

The convergence (2.85) exhibits an interesting phenomenon.

(1) If $u_c/v_c\in\mathbb{R}\setminus\mathbb{Q}$, then for any $\sigma\in\mathbb{R}$, there is a sequence $(k_n)=(k_{1,n}, k_{2,n})\in\mathbb{Z}_+^2$ going to infinity such that $\lim_n k_n/\|k_n\|\to w_c$, for which $\sigma=\lim_n(k_{1,n}-k_{2,n}u_c/v_c)$ and relation (2.85) holds.

(2) The rational case, i.e; when $u_c/v_c=p/q\in\mathbb{Q}$ for some $p, q\in\mathbb{N}^*$, relation (2.85) holds if and only if for $n$ large enough, $qk_{1,n}-k_{2,n}p$ is constant and, therefore, $\sigma\in\mathbb{Q}$ is of the form $\sigma=\tilde{p}/q$ for some $\tilde{p}\in\mathbb{Z}$. An analogous result has been established in Ignatiouk et al. [19].

Case iv) The asymptotics (2.80) reflect a competition between the geometric decay determined by the simple pole $x_d=x^{**}$ of the function $x\mapsto H_j(x, 0)$ and the geometric decay multiplied by a factor $k_2$ determined by the pole $y_d=Y_2(x_d)=y^{**}$ of the second order of the function $y\mapsto H_j(0, y)$. If (B3) and $y_d<y_P^{**}$ hold and $(k_n)=(k_{1,n}, k_{2,n})$ is a sequence of points of $\mathbb{Z}_+^2$ going to infinity such that $\|k_n\|\to+\infty$ and $k_n/\|k_n\|\to(0, 1)$, Theorem 2.4 shows that for $j\in\mathbb{Z}_+^2\setminus E_0$,

$$
\begin{align*}
g(j,k) &\sim \varkappa_1(j) C_1 x_d^{-k_{1,n}-1} y_d^{-k_{2,n}-1} && \text{if } \lim_{n} k_{2,n} x_d^{k_{1,n}} / (X_2(y_d))^{k_{1,n}} = 0; \\
g(j,k) &\sim \varkappa_1(j) C_2 (X_2(y_d))^{-k_{1,n}-1} k_{2,n} y_d^{-k_{2,n}-2} && \text{if } \lim_{n} k_{2,n} x_d^{k_{1,n}} / (X_2(y_d))^{k_{1,n}} = +\infty; \\
g(j,k) &\sim \varkappa_1(j) (C_1 + \sigma C_2 y_d^{-1}) x_d^{-k_{1,n}-1} y_d^{-k_{2,n}-1} && \text{if } \lim_{n} k_{2,n} x_d^{k_{1,n}} / (X_2(y_d))^{k_{1,n}} = \sigma > 0.
\end{align*}
$$

Case v) The asymptotic (2.82) reflects a competition between the geometric decay multiplied by a factor $k_1$ determined by the pole $x_d=X_2(y_d)=x^{**}$ of the second order of the function $x\mapsto H_j(x, 0)$ and the geometric decay determined by the simple pole $y_d=y^{**}$ of the function $y\mapsto H_j(0, y)$.

Similar asymptotics hold, by exchanging the roles of $x$ and $y$ in this case when $x_d<x_P^{**}$, $\|k_n\|\to+\infty$ and $k_n/\|k_n\|\to(1, 0)$.

## Relations with Homogeneous Random Walks.

The asymptotic relation (2.84) is similar to the exact asymptotics of the Green function of the homogeneous random walk $(S(n))$ in $\mathbb{Z}^2$, see Ney and Spitzer [27] and Theorem 25.15 of Woess [30]. The only difference in fact is that in the asymptotic obtained by Ney and Spitzer [27], the function $j=(j_1, j_1)\mapsto(x_D(w_k))^{j_1}(y_D(w_k))^{j_1}=\exp(\alpha_D(w_k)j_1 + \beta_D(w_k)j_2)$, which
is harmonic for the homogeneous random walk, is replaced in our case by the function $j \mapsto \kappa_{(x_D(w_k), y_D(w_k))}(j)$, which is harmonic for the killed random walk $(Z_{\tau_0}(n))$. Analogous results have been obtained in Ignatiouk [18] for the Green functions of a homogeneous random walk in $\mathbb{Z}^d$ killed outside of an open cone and for the asymptotics along the interior directions of the cone.

## Outline of the paper

Our paper is organized as follows:

- In Section 3, our results are used to investigate the Martin compactification of $\mathbb{Z}_+^2$ of the killed Markov chain $(Z_{\tau_0}(n))$ and also of the original random walk $(Z(n))$ when it is transient.

- Section 4 contains the proof of preliminary results. It is shown that the points $x_P^*$, $x^*$, $x^{**}$, $x_P^{**}$, $y_P^*$, $y^*$, $y^{**}$, $y_P^{**}$ and the functions $Y_1$, $Y_2: [x_P^*, x_P^{**}] \to [y_P^*, y_P^{**}]$ and $X_1$, $X_2: [y_P^*, y_P^{**}] \to [x_P^*, x_P^{**}]$ are well defined, and their firs properties are obtained.

- Proposition 2.2 is proved in Section 5.

- Sections 6 - 9 are respectively devoted to the proofs of Theorems 2.1 - 2.6.

- In Section A, the conditions (B0)-(B7) are compared with the conditions of positive recurrence and transience for the random walk $(Z(n))$;

## 3 Application to the Martin Boundary

When the Markov chain $(Z(n))$ is transient, for $j, k \in \mathbb{Z}_+^2$, its Green function

$$G(j, k) = \sum_{n=0}^{\infty} \mathbb{P}_j(Z(n) = k),$$

is related to the Green function $g(j, k)$ of the killed Markov chain at 0 in the following way,

$$G(j, k) = g(j, k) + G(j, 0)g(0, k), \quad \text{for } j \neq 0,$$

and

$$G(0, k) = G(0, 0)g(0, k).$$

The Martin kernels have therefore a simple relation

$$\frac{G(j, k)}{G(0, k)} = \frac{1}{G(0, 0)} \frac{g(j, k)}{g(0, k)} + \frac{G(j, 0)}{G(0, 0)} = \mathbb{P}_0(\tau_0 = +\infty) \frac{g(j, k)}{g(0, k)} + P_j(\tau_0 < +\infty), \quad (3.1)$$

and the Martin compactification of $\mathbb{Z}_+^2$ for the killed Markov chain $(Z(n))$ at 0 is homeomorphic to the Martin compactification of the original random walk $(Z(n))$.

In this section, our asymptotic results, Theorems 2.4 and 2.6, are used to obtain the asymptotics of the Martin kernel of the killed Markov chain $(Z(n))$ at 0. We do not assume that the original random walk $(Z(n))$ is positive recurrent or transient. The Martin boundary of $(g(j, k))$ is in fact almost completely described, the four asymptotic cases mentioned in Section 2.6.3 excepted. Throughout this section, we will assume that the conditions (A1)-(A3) are satisfied.

Since by Lemma 2.2, $g(j, k) = 0$, for any $j \in E_0$, and $k \in \mathbb{Z}_+^2 \setminus E_0$, to investigate the Martin boundary of the killed random walk, it is sufficient to consider $j \in \mathbb{Z}_+^2 \setminus E_0$.
**Proposition 3.1.** Suppose that either (B0) or (B1) holds, and let the vector $w_c=(u_c, v_c) \in \mathbb{S}_+^1$ be the critical direction defined in Section 2.6.2 and $(k_n)=(k_{1,n}, k_{2,n})$ be a sequence of $Z_+^2$ whose norm goes to infinity. Then for any $j \in \mathbb{Z}_+^2 \setminus E_0$, the following assertions hold.

(1) If $\liminf_{n} k_{1,n}/\|k_n\| > u_c$ then
$$ \lim_{n \to +\infty} \frac{g(j, k_n)}{g(0, k_n)} = \frac{\kappa_1(j)}{\kappa_1(0)}, \qquad (3.2) $$

where $\kappa_1$ is the function defined by relation (2.35).

(2) If $\limsup_{n} k_{1,n}/\|k_n\| < u_c$ then
$$ \lim_{n \to +\infty} \frac{g(j, k_n)}{g(0, k_n)} = \frac{\kappa_2(j)}{\kappa_2(0)}, \qquad (3.3) $$

where $\kappa_2$ is the function defined by relation (2.37).

(3) If (B0) holds and $\lim_{n} k_n/\|k_n\| = w_c$, then
$$ \lim_{n \to +\infty} \frac{g(j, k_n)}{g(0, k_n)} = \frac{b_1\kappa_1(j)\lambda_1^{-k_1-1} + b_2\kappa_2(j)\lambda_2^{-k_2-1}}{b_1\kappa_1(0)\lambda_1^{-k_1-1} + b_2\kappa_2(0)\lambda_2^{-k_2-1}}, $$

with $\lambda_1=x_d/X_2(y_d)$ and $\lambda_2=y_d/Y_2(x_d)$ and $b_1>0$, resp. $b_2>0$, is given by relation (2.76), resp. relation (2.78).

When either (B0) or (B1) holds, Proposition 3.1 implies that the minimal Martin boundary $\partial_m \mathbb{Z}_+^2$, see Proposition 24.4 of [30], contains two points $\xi_1$ and $\xi_2$, and any sequence of points $(k_n)$ converging to infinity and such that $\liminf_{n} k_{1,n}^{(n)}/\|k^{(n)}\| > u_c$ (resp. $\limsup_{n} k_{1,n}^{(n)}/\|k^{(n)}\| < u_c$) converges in the Martin compactification of $\mathbb{Z}_+^2$ to $\xi_1$ (resp. to $\xi_2$).

When (B0) holds, with this result one gets that the minimal Martin boundary of $\mathbb{Z}_+^2$ relative for the killed Markov chain at 0 contains exactly two points, and with the same arguments as in Theorem 3 of [19], we obtain that the full Martin boundary is homeomorphic to $\mathbb{Z} \cup \{\pm\infty\}$, resp. $\mathbb{R} \cup \{\pm\infty\}$, if $u_c/v_c \in \mathbb{Q}$, resp. $u_c/v_c \notin \mathbb{Q}$. By the Poisson-Martin representation theorem, in this case, any non-negative harmonic function for the killed Markov chain is therefore of the form $\theta_1\kappa_1 + \theta_2\kappa_2$ with for some $\theta_1, \theta_2 \in [0, +\infty[$.

Note that the full Martin boundary is not obtained in the case (B1) since the asymptotics of the Green function $g(j, k)$ along the direction $w_c$ are missing. See Section 2.6.3.

For the region (B2) we have the following proposition.

**Proposition 3.2.** Suppose that (B2) holds, and let $(k_n)=(k_{1,n}, k_{2,n})$ be a sequence of $Z_+^2$ whose norm goes to infinity. Then for any $j \in \mathbb{Z}_+^2 \setminus E_0$, the following assertions hold.

(1) If $\lim_{n} k_n/\|k_n\| = w \in \mathcal{W}_0$, then
$$ \lim_{n \to +\infty} \frac{g(j, k_n)}{g(0, k_n)} = \frac{\kappa_{(x_D(w), y_D(w))}(j)}{\kappa_{(x_D(w), y_D(w))}(0)} \qquad (3.4) $$

where $\kappa_{(\cdot, \cdot)}$ is the function defined by (2.39).

(2) If $x_d=x_P^{**}$ and the sequence $(k_{2,n})$ is bounded, then

$$ \lim_{n \to +\infty} \frac{g(j, k_n)}{g(0, k_n)} = \begin{cases} \tilde{\kappa}_1(j)/\tilde{\kappa}_1(0) & \text{if } \phi_1(x_P^{**}, Y_1(x_P^{**})) < 1, \\ \kappa_1(j)/\kappa_1(0) & \text{if } \phi_1(x_P^{**}, Y_1(x_P^{**})) = 1. \end{cases} \qquad (3.5) $$

where $\tilde{\kappa}_1$ is the function defined by (2.36).
(3) If $x^{**}<x_P^{**}$ and $\liminf_{n} k_{1,n}/\|k_n\|>u_D(x_d, Y_2(x_d))$, then

$$ \lim_{n \to +\infty} \frac{g(j, k_n)}{g(0, k_n)} = \frac{\kappa_1(j)}{\kappa_1(0)}. $$

(4) if $y^{**}=y_P^{**}$ and the sequence $(k_{1,n})$ is bounded, then

$$ \lim_{n \to +\infty} \frac{g(j, k_n)}{g(0, k_n)} = \begin{cases} \tilde{\kappa}_2(j)/\tilde{\kappa}_2(0) & \text{if } \phi_2(X_1(y_P^{**}), y_P^{**}) < 1, \\ \kappa_2(j)/\kappa_2(0) & \text{if } \phi_2(X_1(y_P^{**}), y_P^{**}) = 1 \end{cases} \quad (3.6) $$

where $\tilde{\kappa}_2$ is the function defined by (2.38).

(5) If $y^{**}<y_P^{**}$ and $\liminf_{n} k_{2,n}/\|k_n\|>v_D(x_d, Y_2(x_d))$ then,

$$ \lim_{n \to +\infty} \frac{g(j, k_n)}{g(0, k_n)} = \frac{\kappa_2(j)}{\kappa_2(0)}. $$

When (B2) holds, this result proves that for any direction $w \in S_+^1$ in the closure $\overline{W}_0$ of $W_0$ there is a point $\xi(w)$ in the Martin boundary of the killed Markov chain, and that if $(k_n)$ is a sequence of points of $\mathbb{Z}_+^2$ whose norm converges to infinity then, for the convergence in the Martin compactification,

$$ \lim_{n \to +\infty} k_n = \begin{cases} \xi(w) & \text{for } w \in W_0 \text{ if } k_n/\|k_n\| \to w, \\ \xi(w_D(x_d, Y_2(x_d))) & \text{if } \liminf_{n} k_{1,n}/\|k_n\| > u_D(x_d, Y_2(x_d)), \\ \xi(w_D(X_2(y_d), y_d)) & \text{if } \limsup_{n} k_{1,n}/\|k_n\| < u_D(X_2(y_d), y_d). \end{cases} $$

As explained before, in the case (B2), also due to the missing asymptotics for the singular directions $w_D(x_d, Y_2(x_d))$ and $w_D(X_2(y_d), y_d)$, the full Martin boundary is not completely determined. We conjecture that in this case it is homeomorphic to $\overline{W}_0$.

**Proposition 3.3.** Suppose that either (B3) and $y_d<y_P^{**}$ hold or (B4) holds. Then for any $j \in \mathbb{Z}_+^2 \setminus E_0$,

$$ \lim_{\|k\| \to +\infty} \frac{g(j, k)}{g(0, k)} = \frac{\kappa_1(j)}{\kappa_1(0)}, $$

If either (B3) and $y_d<y_P^{**}$ hold or (B4) holds, the full Martin boundary of $\mathbb{Z}_+^2$ of the killed Markov chain is therefore a single point, and, up to a multiplicative constant, the function $\kappa_1$ is the unique non-negative harmonic function.

When the original random walk $(Z(n))$ is transient and the measures $\mu_0$, $\mu_1$ and $\mu_2$ are stochastic, since the Martin compactification of $\mathbb{Z}_+^2$ of the Markov chain $(Z(n))$ is homeomorphic to the Martin compactification of $\mathbb{Z}_+^2$ of the killed Markov chain, we have the following corollary.

**Corollary 3.4.** If the measures $\mu_0$, $\mu_1$ and $\mu_2$ are stochastic and the Markov chain $(Z(n))$ is transient, then under the hypotheses of Proposition 3.3, for any $j \in \mathbb{Z}_+^2$,

$$ \lim_{\|k\| \to +\infty} \frac{G(j, k)}{G(0, k)} = 1. \quad (3.7) $$

*Proof.* Indeed, in this case, using (3.1), Proposition 3.3 and Lemma 2.2, since the set $E_0$ does not contain the origin, one gets

$$ \lim_{\|k\| \to +\infty} \frac{G(j, k)}{G(0, k)} = \mathbb{P}_0(\tau_0 = +\infty)\kappa_1(j) + P_j(\tau_0 < +\infty), \quad \forall j \in \mathbb{Z}_+^2, $$
and consequently, by the Poisson-Martin representation theorem, up to a multiplicative constant, for any harmonic function $\varkappa$ for the Markov chain $(Z(n))$, one has

$$ \varkappa(j)/\varkappa(0) = \mathbb{P}_0(\tau_0 = +\infty)\varkappa_1(j) + P_j(\tau_0 < +\infty), \quad \forall j \in \mathbb{Z}_+^2. $$

Since in the case when the measures $\mu_0$, $\mu_1$ and $\mu_2$ are stochastic, the constants functions are harmonic for $(Z(n))$, from this it follows that

$$ \mathbb{P}_0(\tau_0 = +\infty)\varkappa_1(j) + P_j(\tau_0 < +\infty) = 1, \quad \forall j \in \mathbb{Z}_+^2, \qquad (3.8) $$

and (3.7) holds.

Remark that the identity (3.8) can also be obtained in a straightforward way by using Proposition A.3 of Section A. $\square$

By symmetry, similar result can be obtained with $\varkappa_2$ instead of $\varkappa_1$ if either (B5) and $x_d = x_P^{**}$ hold or (B6) holds.

We conclude with the case when either (B3) holds with $y_d=y_P^{**}$ or (B5) holds with $x_d=x_P^{**}$.

**Proposition 3.5.** Suppose that (B3) holds with $y_d=y_P^{**}$ and let $(k_n)$ be a sequence of points of $\mathbb{Z}_+^2$ whose norm converges to infinity. Then for any $j \in \mathbb{Z}_+^2 \setminus E_0$, the convergence

$$ \lim_{n \to +\infty} \frac{g(j, k_n)}{g(0, k_n)} = \frac{\varkappa_1(j)}{\varkappa_1(0)} $$

holds in any of the two following cases:

- when $\liminf_{n} k_{1,n}/\|k_n\|>0$;

- when the sequence $(k_{1,n})$ is bounded and $\lim_{n} k_{2,n} = +\infty$.

When (B3) holds with $y_d=y_P^{**}$ there is therefore a point $\xi_0$ in the Martin boundary of $\mathbb{Z}_+^2$ of the killed Markov chain, such that any sequence of points of $\mathbb{Z}_+^2$ satisfying the conditions of Proposition 3.5 converges in the Martin compactification to $\xi_0$.

Similar result with the function $\varkappa_2$ instead of $\varkappa_1$ can be obtained when (B5) holds with $x_d=x_P^{**}$.

When (B3) holds with $y_d=y_P^{**}$, resp. (B5) holds with $x_d=x_P^{**}$, the asymptotics of the Green function $g(j,k)$ when $\|k\| \to +\infty$ and $k_1/\|k\| \to 0$, resp. when $\|k\| \to +\infty$ and $k_2/\|k\| \to 0$ are not known. See Section 2.6.3. For this reason, the full Martin boundary is not determined in this case. We conjecture it is a single point.

# 4 Preliminary results

In the following statement, we investigate the set $D$ defined by (2.7): it is proved that the line segments $[x_P^*, x_P^{**}]$, $[y_P^*, y_P^{**}]$ and the functions $Y_1, Y_2: [x_P^*, x_P^{**}] \to [y_P^*, y_P^{**}]$ and $X_1, X_2: [y_P^*, y_P^{**}] \to [x_P^*, x_P^{**}]$ are well defined, and the first useful for our purpose properties of these functions are obtained.

**Lemma 4.1.** Under the hypotheses (A1), the line segments $[x_P^*, x_P^{**}]$ and $[y_P^*, y_P^{**}]$ and the functions $Y_1, Y_2: [x_P^*, x_P^{**}] \to [y_P^*, y_P^{**}]$ and $X_1, X_2: [y_P^*, y_P^{**}] \to [x_P^*, x_P^{**}]$ are well defined and the following assertions hold:

1) For any $\hat{x} \in [x_P^*, x_P^{**}]$, $Y_1(\hat{x})$ and $Y_2(\hat{x})$ are the only real positive solutions of the equation $P(\hat{x}, y) = 1$,

$$ Y_1(\hat{x}) < Y_2(\hat{x}), \quad \partial_y P(\hat{x}, Y_1(\hat{x})) < 0 \quad \text{and} \quad \partial_y P(\hat{x}, Y_2(\hat{x})) < 0 \quad \text{if} \quad x_P^* < \hat{x} < x_P^{**}, $$

and

$$ Y_1(\hat{x}) = Y_2(\hat{x}) \quad \text{and} \quad \partial_y P(\hat{x}, Y_1(\hat{x})) = 0 \quad \text{if} \quad \hat{x} \in \{x_P^*, x_P^{**}\}. $$
2) For any $\hat{y} \in [y_P^*, y_P^{**}]$, $X_1(\hat{y})$ and $X_2(\hat{y})$ are the only real positive solutions of the equation
$P(x, \hat{y}) = 1$,

$$X_1(\hat{y}) < X_2(\hat{y}), \quad \partial_x P(X_1(\hat{y}), \hat{y}) < 0 \quad \text{and} \quad \partial_x P(X_2(\hat{y}), \hat{y}) \quad \text{if} \quad y_P^* < \hat{y} < y_P^{**},$$

and

$$X_1(\hat{y}) = X_2(\hat{y}) \quad \text{and} \quad \partial_x P(X_1(\hat{y}), \hat{y}) = 0 \quad \text{if} \quad \hat{y} \in \{y_P^*, y_P^{**}\}.$$

3) The four points $(x_P^*, Y_1(x_P^*))$, $(x_P^{**}, Y_1(x_P^{**}))$, $(X_1(y_P^*), y_P^*)$ and $(X_1(y_P^{**}), y_P^{**})$ are two by two distinct and moreover $Y_1(x_P^*), Y_1(x_P^{**}) \in [y_P^*, y_P^{**}]$ and $X_1(y_P^*), X_1(y_P^{**}) \in [x_P^{**}, x_P^{**}]$.

4) The function $X_1 : [y_P^*, Y_1(x_P^*)] \to [x_P^*, X_1(y_P^*)]$ is strictly decreasing and its inverse is
$Y_1 : [x_P^*, X_1(y_P^*)] \to [y_P^*, Y_1(x_P^*)]$.

5) The function $X_1 : [Y_1(x_P^*), y_P^{**}] \to [x_P^*, X_1(y_P^{**})]$ is strictly increasing and its inverse is
$Y_2 : [x_P^*, X_1(y_P^{**})] \to [Y_1(x_P^*), y_P^{**}]$.

6) The function $X_2 : [y_P^*, Y_1(x_P^{**})] \to [X_1(y_P^*), x_P^{**}]$ is strictly increasing and its inverse is
$Y_1 : [X_1(y_P^*), x_P^{**}] \to [y_P^*, Y_1(x_P^{**})]$.

7) The function $X_2 : [Y_1(x_P^{**}), y_P^{**}] \to [X_1(y_P^{**}), x_P^{**}]$ strictly decreasing and its inverse is
$Y_2 : [X_1(y_P^{**}), x_P^{**}] \to [Y_1(x_P^{**}), y_P^{**}]$.

**Proof.** To prove this lemma we notice that under the hypotheses (A1) the jump generating function $\tilde{P} : \mathbb{R}^2 \to \mathbb{R}_+$ defined by (2.63) is strictly convex and finite in a neighborhood of the set $\tilde{D} = \{(\alpha, \beta) \in \mathbb{R}^2 : \tilde{P}(\alpha, \beta) \le 1\}$. The set $\tilde{D}$ is therefore strictly convex and compact (see for instance, Spitzer [29]) and because of the assumption (A1) (iii), it has a non-empty interior. Since the mapping $(\alpha, \beta) \mapsto (x, y)=(e^\alpha, e^\beta)$ determines a homeomorphism from $\mathbb{R}^2$ to $]0, +\infty[$ and maps the set $\tilde{D}$ to the set $D$, one gets that $x_P^* = e^{\alpha_P^*}, x_P^{**} = e^{\alpha_P^{**}}, y_P^* = e^{\beta_P^*}$ and $y_P^{**} = e^{\beta_P^{**}}$ where

$$[\alpha_P^*, \alpha_P^{**}] = \{\alpha \in \mathbb{R} : \inf_{\beta \in \mathbb{R}} \tilde{P}(\alpha, \beta) \le 1\} \quad [\beta_P^*, \beta_P^{**}] = \{\beta \in \mathbb{R} : \inf_{\alpha \in \mathbb{R}} \tilde{P}(\alpha, \beta) \le 1\}.$$

Our lemma holds therefore with $Y_1(\hat{x}) = e^{\beta_1(\ln \hat{x})}$, $Y_2(\hat{x}) = e^{\beta_2(\ln \hat{x})}$, $X_1(\hat{y}) = e^{\alpha_1(\ln \hat{y})}$ and $X_2(\hat{y}) = e^{\alpha_2(\ln \hat{y})}$ where for $\hat{\alpha} \in [\alpha_P^*, \alpha_P^{**}]$, $\beta_1(\hat{\alpha}) \le \beta_2(\hat{\alpha})$ are the only real solutions of the equation $\tilde{P}(\hat{\alpha}, \beta) = 1$ and for $\hat{\beta} \in [\beta_P^*, \beta_P^{**}] \le \alpha_1(\hat{\beta})$ and $\alpha_2(\hat{\beta})$ are the only real solutions of the equation $\tilde{P}(\hat{\alpha}, \hat{\beta}) = 1$. $\square$

In the following lemma, we investigate the sets $D \cap D_1$ and $D \cap D_2$. It is proved that the line segments $[x^*, x^{**}]$ and $[y^*, y^{**}]$ are well defined. With this result, we will be able to get the first useful for our purpose properties of the functions $x \mapsto \phi_1(x, Y_1(x))$ and $y \mapsto \phi_2(X_1(y), y)$.

**Lemma 4.2.** Under the hypotheses (A1) and (A3), there exist $x^*, x^{**} \in ]0, +\infty[$ such that $x^* < x^{**}$ and

$$[x^*, x^{**}] = \{x \in ]0, +\infty[ : (x, y) \in D_1 \cap D \text{ for some } y > 0\}.$$

Moreover, for any $x \in [x^*, x^{**}]$, there exists $\tilde{Y}_2(x) \ge Y_1(x)$ such that

$$\{y > 0 : (x, y) \in D_1 \cap D\} = [Y_1(x), \tilde{Y}_2(x)],$$

the points $Y_1(x)$ and $\tilde{Y}_2(x)$ are the only real positive solutions of the equation

$$\max\{P(x, y), \phi_1(x, y)\} = 1, \quad (4.1)$$

and $Y_1(x) < \tilde{Y}_2(x)$ if and only if $x^* < x < x^{**}$.
Proof. To prove this lemma, we consider the jump generating function $\tilde{P}$ defined by (2.63), the jump generating function $\tilde{\phi}_1 : \mathbb{R}^2 \to \mathbb{R}_+$, defined for $(\alpha, \beta) \in \mathbb{R}^2$, by

$$ \tilde{\phi}_1(\alpha, \beta) = \sum_{k=(k_1, k_2) \in \mathbb{Z}^2} \exp(\alpha k_1 + \beta k_2) \mu_1(k) $$

and the sets $\tilde{D} = \{(\alpha, \beta) \in \mathbb{R}^2 : \tilde{P}(\alpha, \beta) \le 1\}$ and $\tilde{D}_1 = \{(\alpha, \beta) \in \mathbb{R}^2 : \tilde{\phi}_1(\alpha, \beta) \le 1\}$. Recall that for $x = e^\alpha$ and $y = e^\beta$,

$$ \tilde{P}(\alpha, \beta) = P(x, y), \quad \tilde{\phi}_1(\alpha, \beta) = \phi_1(x, y), $$

and that the mapping $(\alpha, \beta) \mapsto (x, y) = (e^\alpha, e^\beta)$ determines a homeomorphism from $\mathbb{R}^2$ to $]0, +\infty]^2$ and maps the set $\tilde{D} \cap \tilde{D}_1$ to the set $D \cap D_1$. Hence, to prove our lemma, it is sufficient to show that under our assumptions, the following assertions hold

- there exist $\alpha^*, \alpha^{**} \in \mathbb{R}$ such that $\alpha^* < \alpha^{**}$ and

$$ [\alpha^*, \alpha^{**}] = \{\alpha \in \mathbb{R} : (\alpha, \beta) \in \tilde{D}_1 \cap \tilde{D} \text{ for some } \beta \in \mathbb{R}\}. $$

- for any $\alpha \in [\alpha^*, \alpha^{**}]$, there exists $\tilde{\beta}_2(\alpha) \ge \ln Y_1(e^\alpha)$ such that

$$ \{\beta \in \mathbb{R} : (\alpha, \beta) \in \tilde{D}_1 \cap \tilde{D}\} = [\ln Y_1(e^\alpha), \tilde{\beta}_2(\alpha)], $$

the points $\beta_1(\alpha) = \ln Y_1(e^\alpha)$ and $\tilde{\beta}_2(\alpha)$ are the only real positive solutions of the equation

$$ \max\{\tilde{P}(\alpha, \beta), \tilde{\phi}_1(\alpha, \beta)\} = 1, \quad (4.2) $$

and $\ln Y_1(e^\alpha) < \tilde{\beta}_2(\alpha)$ if and only if $\alpha^* < \alpha < \alpha^{**}$.

For this we remark that the set $\tilde{D} \cap \tilde{D}_1$ is compact convex and has a non empty interior because

- the function $\tilde{\phi}_1$ is convex and consequently also the set $\tilde{D}_1$ is also convex;

- the set $\tilde{D}$ is convex and compact (see the proof of Lemma 4.1);

- by Assumption (A3)(iii), the set $D \cap D_1$ has a non-empty interior, and consequently, since the mapping $(\alpha, \beta) \mapsto (x, y) = (e^\alpha, e^\beta)$ determines a homeomorphism from $\mathbb{R}^2$ to $]0, +\infty]^2$ and maps the set $\tilde{D}_1 \cap \tilde{D}$ to the set $D_1 \cap D$, the set $\tilde{D}_1 \cap \tilde{D}$ has also a non-empty interior.

From this, it follows that the following assertions hold:

- there exist $\alpha^* > 0$ and $\alpha^{**} > \alpha^*$ such that

$$ [\alpha^*, \alpha^{**}] = \{\alpha \in \mathbb{R} : (\alpha, \beta) \in \tilde{D}_1 \cap \tilde{D} \text{ for some } \beta \in \mathbb{R}\}, $$

- for any $\alpha \in [\alpha^*, \alpha^{**}]$, there exists $\tilde{\beta}_1(\alpha), \tilde{\beta}_2(\alpha) \in ]0, +\infty[$ such that $\tilde{\beta}_1(\alpha) \le \tilde{\beta}_2(\alpha)$,

$$ [\tilde{\beta}_1(\alpha), \tilde{\beta}_2(\alpha)] = \{\beta \in \mathbb{R} : (\alpha, \beta) \in \tilde{D} \cap \tilde{D}_1\}, $$

and

$$ \max\{\tilde{\phi}_1(\alpha, \tilde{\beta}_1(\alpha)), \tilde{P}(\alpha, \tilde{\beta}_1(\alpha))\} = \max\{\tilde{\phi}_1(\alpha, \tilde{\beta}_2(\alpha)), \tilde{P}(\alpha, \tilde{\beta}_2(\alpha))\} = 1, $$

- for any $\alpha \in ]\alpha^*, \alpha^{**}[$

$$ \tilde{\beta}_1(\alpha) < \tilde{\beta}_2(\alpha). $$
Hence to complete our proof, it is sufficient to show that for any $\alpha \in [\alpha^*, \alpha^{**}]$, the following relations hold:

$$ \tilde{\beta}_1(\alpha) = \ln Y_1(e^{\alpha}), \qquad (4.3) $$

$$ \max\{\tilde{P}(\alpha, \beta), \tilde{\phi}_1(\alpha, \beta)\} < 1 \quad \text{if} \quad \beta_1(\alpha) < \beta < \tilde{\beta}_2(\alpha), \qquad (4.4) $$

and

$$ \alpha^* < \alpha < \alpha^{**} \quad \text{if} \quad \tilde{\beta}_1(\alpha) < \tilde{\beta}_2(\alpha). \qquad (4.5) $$

Remark moreover that if (4.4) holds, then any $\beta$ such that $\tilde{\beta}_1(\alpha) < \beta < \tilde{\beta}_2(\alpha)$, the point $(\alpha, \beta)$ belongs to the interior of the set $\tilde{D} \cap \tilde{D}_1$ and consequently, (4.5) also holds. Hence, to complete our proof, it is sufficient to prove that for any $\alpha \in [\alpha^*, \alpha^{**}]$, (4.3) and (4.4) hold. To get these relations, we remark that by Lemma 4.1 and since $\tilde{D} \cap \tilde{D}_1 \subset \tilde{D}$, one has

$$ [\alpha^*, \alpha^{**}] \subset \{\alpha \in \mathbb{R} : (\alpha, \beta) \in \tilde{D} \quad \text{for some} \quad \beta \in \mathbb{R}\} = [\ln x_P^*, \ln x_P^{**}] $$

and that for any $\alpha \in [\alpha^*, \alpha^{**}]$,

$$ [\tilde{\beta}_1(\alpha), \tilde{\beta}_2(\alpha)] \subset \{\beta \in \mathbb{R} : (\alpha, \beta) \in \tilde{D}\} = [\ln Y_1(e^{\alpha}), \ln Y_2(e^{\alpha})]. $$

Since because of Assumption (A3), the function $\beta \mapsto \tilde{\phi}_1(\alpha, \beta)$ is finite and strictly increasing in a neighborhood of the line segment $[0, \ln Y_2(e^{\alpha})]$, one gets therefore that for any $\alpha \in [\alpha^*, \alpha^{**}]$ and $\beta < \tilde{\beta}_2(\alpha)$,

$$ \tilde{\phi}_1(\alpha, \beta) < \tilde{\phi}_1(\alpha, \tilde{\beta}_2(\alpha)) \le 1. $$

This implies that for any $\alpha \in [\alpha^*, \alpha^{**}]$, (4.3) holds. Moreover, since by Lemma 4.1, $\tilde{P}(\alpha, \beta) < 1$ for any $\beta \in] \ln Y_1(e^{\alpha}), \ln Y_2(e^{\alpha})[$, from the last relation it follows that for any $\beta \in] \ln Y_1(e^{\alpha}), \tilde{\beta}_2(\alpha)[$, (4.4) also holds. $\square$

As a consequence of Lemma 4.2 we obtain

**Corollary 4.1.** Under the assumptions (A1) and (A3), for any $x \in [x_P^*, x_P^{**}]$,

(i) $x \in [x^*, x^{**}]$ if and only if $\phi_1(x, Y_1(x)) \le 1$;

(ii) $\phi_1(x, Y_1(x)) < 1$ if $x^* < x < x^{**}$;

(iii) $\phi_1(x^{**}, Y_1(x^{**})) = 1$ if $x^{**} < x_P^{**}$;

(iv) $\phi_1(x^*, Y_1(x^*)) = 1$ if $x^* > x_P^*$.

*Proof.* Indeed, since $P(x, Y_1(x)) = 1$ for any $x \in [x_P^*, x_P^{**}]$, the first assertion of this corollary follows from Lemma 4.2.

To get (ii), we recall that by Lemma 4.2, for any $x^* < x < x^{**}$ one has $Y_1(x) < \tilde{Y}_2(x)$ and $\max\{\phi_1(x, \tilde{Y}_2(x)), P(x, \tilde{Y}_2(x))\} = 1$. Since under the hypotheses (A3), for any $x \in [x^*, P, x_P^{**}]$, the function $y \mapsto \phi_1(x, y)$ is finite and strictly increasing in a neighborhood of the line segment $[0, \ln Y_2(e^{\alpha})]$, from this it follows that $\phi_1(x, Y_1(x)) < 1$ whenever $x^* < x < x^{**}$. The assertion (ii) is therefore also proved.

Suppose now that $x^{**} < x_P^{**}$. Then by the first assertion of our corollary,

$$ \phi_1(x, Y_1(x)) > 1 \quad \text{if} \quad x^{**} < x \le x_P^{**}, \qquad (4.6) $$

and by the second assertion,

$$ \phi_1(x, Y_1(x)) < 1 \quad \text{if} \quad x^* < x < x^{**}. \qquad (4.7) $$

Since the function $\phi_1$ is continuous in a neighborhood of the set $D$, the function $Y_1$ is continuous on $[x_P^*, x_P^{**}]$ and $(x, Y_1(x)) \in D$ for any $x \in [x_P^*, x_P^{**}]$, the function $x \mapsto \phi_1(x, Y_1(x))$ is therefore continuous on $[x_P^*, x_P^{**}]$, and consequently, relations (4.6) and (4.7) prove that $\phi_1(x^{**}, Y_1(x^{**})) = 1$. The assertion (iii) is therefore also proved. The proof of the assertion (iv) is quite similar. $\square$
# 5 The Classification in Eight Regions

This section is devoted to the proof of Proposition 2.2 establishing our classification into eight regions in the set of parameters of the random walk $(Z(n))$.

## 5.1 The main idea of the proof

Remark first of all that the cases (B0)-(B7) have a simple geometrical interpretation: if we denote by $[(x, y), (\tilde{x}, \tilde{y})]$ the line segment in $\mathbb{R}^2$ with the end-points at $(x, y)$ and $(\tilde{x}, \tilde{y})$, then by Lemma 5.1 below, for any $x \in [x_P^*, x_P^{**}]$ and $y \in [y_P^*, y_P^{**}]$, one and only one of the following assertions holds:

- the line segments $[(x, Y_1(x)), (x, Y_2(x))]$ and $[(X_1(y), y), (X_2(y), y)]$ have the common point $(x, y) \in D$,

- these line segments are disjoint and $(x, y) \notin D$.

The case (B0) occurs when the line segments

$$[(x^{**}, Y_1(x^{**})), (x^{**}, Y_2(x^{**}))] \quad \text{and} \quad [(X_1(y^{**}), y^{**}), (X_2(y^{**}), y^{**})] \quad (5.1)$$

have the common point $(x^{**}, y^{**})$ in the interior of the set $D$. The cases (B1), (B3), (B5) and (B7) occur when these line segments have the common point $(x^{**}, y^{**})$ on the boundary of the set $D$:

$$(x^{**}, y^{**}) \in \begin{cases} S_{22} & \text{in the case (B1)}, \\ S_{12} & \text{in the case (B3)}, \\ S_{21} & \text{in the case (B5)}, \\ S_{11} & \text{in the case (B7)}. \end{cases}$$

The cases (B2), (B4) and (B6) occur when the line segments (5.1) are disjoint and $(x^{**}, y^{**}) \notin D$. The nearest to $(x^{**}, y^{**})$ vertices of these line segments belong to $S_{22}$ in the case (B2), to $S_{12}$ in the case (B4) and to $S_{21}$ in the case (B6).

The main idea of our proof is the following:

First, we show that the point $(x^{**}, y^{**})$ either belongs to each of the line segments

$$[(x^{**}, Y_1(x^{**})), (x^{**}, Y_2(x^{**}))] \quad \text{and} \quad [(X_1(y^{**}), y^{**}), (X_2(y^{**}), y^{**})] \quad (5.2)$$

or does not belong to any of them. With this result we describe all possible cases:

$$
\begin{align*}
(a) \quad & X_1(y^{**}) < x^{**} < X_2(y^{**}) \quad \text{and} \quad Y_1(x^{**}) < y^{**} < Y_2(x^{**}) \\
(b) \quad & X_1(y^{**}) < x^{**} = X_2(y^{**}) \quad \text{and} \quad Y_1(x^{**}) < y^{**} = Y_2(x^{**}) \\
(c) \quad & x^{**} > X_2(y^{**}) \quad \text{and} \quad y^{**} > Y_2(x^{**}) \\
(d) \quad & x^{**} = X_1(y^{**}) \quad \text{and} \quad Y_1(x^{**}) < y^{**} = Y_2(x^{**}) \\
(e) \quad & x^{**} < X_1(y^{**}) \quad \text{and} \quad y^{**} > Y_2(x^{**}) \\
(f) \quad & X_1(y^{**}) < x^{**} = X_2(y^{**}) \quad \text{and} \quad y^{**} = Y_1(x^{**}) \\
(g) \quad & x^{**} > X_2(y^{**}) \quad \text{and} \quad y^{**} < Y_1(x^{**}) \\
(h) \quad & x^{**} = X_1(y^{**}) \quad \text{and} \quad y^{**} = Y_1(x^{**}) \\
(i) \quad & x^{**} < X_1(y^{**}) \quad \text{and} \quad y^{**} < Y_1(x^{**})
\end{align*}
\tag{5.3}
$$

Next, using (2.10), we prove that the cases (B0) - (B7) correspond respectively to the cases (a)-(h), and the case i) never holds.
## 5.2 Preliminary results for the proof of Proposition 2.2

We begin our proof with the following preliminary results.

**Lemma 5.1.** Under the hypotheses (A1), for any $x \in [x_P^*, x_P^{**}]$ and $y \in [y_P^*, y_P^{**}]$, the line segments $[(x, Y_1(x)), (x, Y_2(x))]$ and $[(X_1(y), y), (X_2(y), y)]$ are disjoint if and only if the point $(x, y)$ does not belong to the set $D$.

*Proof.* Indeed, by Lemma 4.1, for any $x \in [x_P^*, x_P^{**}]$ and $y \in [y_P^*, y_P^{**}]$, the line segment $[(x, Y_1(x)), (x, Y_2(x))]$ is the set of all points $(x', y') \in D$ with $x' = x$, and similarly the line segment $[(X_1(y), y), (X_2(y), y)]$ is the set of all points $(x', y') \in D$ with $y' = y$. Hence, if $(x, y) \in D$ then the both line segments $[(x, Y_1(x)), (x, Y_2(x))]$ and $[(X_1(y), y), (X_2(y), y)]$ contain the point $(x, y)$. Conversely, if these line segments are not disjoint, then the point $(x, y)$ belongs to each of them, and consequently $(x, y) \in D$. $\square$

Since for any $x \in [x_P^*, x_P^{**}]$ and $y \in [y_P^*, y_P^{**}]$, the point $(x, y)$ is the only point that could belong to the both line segments $[(x, Y_1(x)), (x, Y_2(x))]$ and $[(X_1(y), y), (X_2(y), y)]$, the above lemma implies the following statement.

**Corollary 5.1.** Under the hypotheses (A1), for any $x \in [x_P^*, x_P^{**}]$ and $y \in [y_P^*, y_P^{**}]$, if the point $(x, y)$ does not belong to some of the line segments $[(x, Y_1(x)), (x, Y_2(x))]$ or $[(X_1(y), y), (X_2(y), y)]$ then $(x, y)$ neither belongs to any of them.

By lemma 4.1, for any $x \in [x_P^*, x_P^{**}]$, the points $Y_1(x)$ and $(Y_2(x)$ are the only real and positive solutions of the equation $P(x, y) = 1$, and that $P(x, y) < 1$ for any $y \in]Y_1(x), Y_2(x)]$. Hence, for any $x \in [x_P^*, x_P^{**}]$, each of the points $(x, Y_1(x))$ and $(x, Y_2(x))$ belongs to the boundary of the set $D$, and for any $y \in]Y_1(x), Y_2(x)]$, the point $(x, y)$ belongs to the interior $\mathring{D}$ of $D$. Similarly, for any $y \in [y_P^*, y_P^{**}]$, each of the points $(X_1(y), y)$ and $(X_2(y), y)$ belongs to the boundary of $D$ and for any $x \in]X_1(y), X_2(y)]$, the point $(x, y)$ belongs to $\mathring{D}$. Using Corollary 5.1 it follows the following useful for our purpose property of the set $D$:

**Corollary 5.2.** Under the hypotheses (A1), for any $(x, y) \in D$, one and only one of the following assertions holds

- the point $(x, y)$ belongs to the interior of the set $D$ and in this case, $X_1(y) < x < X_2(y)$ and $Y_1(x) < y < Y_2(x)$;

- the point $(x, y)$ belongs to the boundary of $D$ and in this case, the point $(x, y)$ is an end point of each of the line segments $[(x, Y_1(x)), (x, Y_2(x))]$ and $[(X_1(y), y), (X_2(y), y)]$.

By Lemma 4.1, for any $x \in [x_P^*, x_P^{**}]$, we have

$$Y_1(x) = Y_2(x) \quad \text{if and only if} \quad x \in \{x_P^*, x_P^{**}\}$$

and, for any $y \in [y_P^*, y_P^{**}]$,

$$X_1(y) = X_2(y) \quad \text{if and only if} \quad y \in \{y_P^*, y_P^{**}\},$$

and, the four points $(x_P^*, Y_1(x_P^*))$, $(x_P^{**}, Y_1(x_P^{**}))$, $(X_1(y_P^*), y_P^*)$ and $(X_2(y_P^{**}), y_P^{**})$ are two by two distinct. Hence, the case with $x = X_1(y) = X_2(y)$ and $y = Y_1(x) = Y_2(x)$ is not possible, and as a straightforward consequence of Corollary 5.1 and Corollary 5.2 one gets
**Corollary 5.3.** Under the hypotheses (A1), for any $x \in [x_P^*, x_P^{**}]$ and $y \in [y_P^*, y_P^{**}]$, one and only one of the following cases holds:

$$
\begin{align*}
(a) \quad & X_1(y) < x < X_2(y) \quad \text{and} \quad Y_1(x) < y < Y_2(x) \\
(b) \quad & x = X_2(y) > X_1(y) \quad \text{and} \quad y = Y_2(x) > Y_2(x) \\
(c) \quad & x > X_2(y) \quad \text{and} \quad y > Y_2(x) \\
(d) \quad & x = X_1(y) \quad \text{and} \quad Y_1(x) < y = Y_2(x) \\
(e) \quad & x < X_1(y) \quad \text{and} \quad y > Y_2(x) \\
(f) \quad & X_1(y) < x = X_2(y) \quad \text{and} \quad y = Y_1(x) \\
(g) \quad & x > X_2(y) \quad \text{and} \quad y < Y_1(x) \\
(h) \quad & x = X_1(y) \quad \text{and} \quad y = Y_1(x) \\
(i) \quad & x < X_1(y) \quad \text{and} \quad y < Y_1(x)
\end{align*}
\tag{5.4}
$$

In the case (a) of this statement, the point $(x, y)$ belongs to the interior of the set $D$, in each of the cases (b), (d), (f) and (h), the point $(x, y)$ belongs to the boundary of $D$, and in each of the cases (c), (e), (g) and (i), the point $(x, y)$ does not belong to the set $D$.

When applied with $x = x^{**}$ and $y = y^{**}$, this statement proves that under the hypotheses (A1), one of the assertions (a)-(i) of (5.3) holds.

The following statement will be used to investigate the position of the nearest to $(x^{**}, y^{**})$ vertices of the line segments (5.2) when the point $(x^{**}, y^{**})$ does not belong to the interior of $D$.

**Lemma 5.2.** Under the hypotheses (A1), for any $x \in [x_P^*, x_P^{**}]$ and $y \in [y_P^*, y_P^{**}]$, the following assertions hold :

1) If $x \le X_1(y)$ and $Y_2(x) \le y$, then $(x, Y_2(x)), (X_1(y), y) \in S_{12}$.

2) If $y \le Y_1(x)$ and $X_2(y) \le x$, then $(X_2(y), y), (x, Y_1(x)) \in S_{21}$.

3) If $x \ge X_2(y)$ and $y \ge X_2(x)$ then $(x, Y_2(x)), (X_2(y), y) \in S_{22}$.

4) If $x \le X_1(y)$ and $y \le Y_1(x)$ then $(x, Y_1(x)), (X_1(y), y) \in S_{11}$.

**Proof.** Suppose that $x \in [x_P^*, x_P^{**}]$ and $y \in [y_P^*, y_P^{**}]$ and let $x \le X_1(y)$ and $Y_2(x) \le y$. Then by Corollary 5.3, either $x < X_1(y)$ and $Y_2(x) < y$, or $x = X_1(y)$ and $Y_2(x) = y$. In the second case, i.e when $x = X_1(y)$ and $Y_2(x) = y$, we have $(x, Y_2(x)) = (X_1(y), y)$ and consequently, by the definition of $S_{12}$,

$$ (x, Y_2(x)) = (X_1(y), y) \in S_{12}. $$

Consider now the case when $x < X_1(y)$ and $Y_2(x) < y$. By the definition of the curves (2.11) - (2.14), we have

$$ (x, Y_2(x)) \in S_{12} \cup S_{22} \quad \text{and} \quad (X_1(y), y) \in S_{11} \cup S_{12}. $$

If we suppose that $(x, Y_2(x)) \in S_{22}$, then we will get $Y_2(x) \in [Y_1(x_P^{**}), y_P^{**}]$ and consequently, since $Y_2(x) < y \le y_P^{**}$, we will have also $y \in [Y_1(x_P^{**}), y_P^{**}]$. Since the function $Y_2$ is strictly decreasing on $[Y_1(x_P^{**}), y_P^{**}]$ and since $y > Y_2(x)$, it follows that

$$ X_2(x) < X_2 \circ Y_2(x) = x \tag{5.5} $$

where the last relation holds because by Lemma 4.1, the function $X_2 : [Y_1(x_P^{**}), y_P^{**}] \to [X_2(y_P^{**}), x_P^{**}]$ is inverse to the function $Y_2 : [X_2(y_P^{**}), x_P^{**}] \to [Y_1(x_P^{**}), y_P^{**}]$. Since $X_1(y) \le X_2(y)$, (5.5) contradicts the inequality $x < X_1(y)$ and consequently, when $x < X_1(y)$ and $Y_2(x) < y$, the point $(x, Y_2(x))$ belongs to $S_{12}$. Similar arguments show that in this case, the point $(X_1(y), y)$ also belongs to $S_{12}$. The first assertion of Lemma5.2 is therefore proved. The proof of the assertions 2)-4) is quite similar. $\square$

To investigate the cases (d) and (e) we will need the following preliminary result.
**Lemma 5.3.** If the conditions (A1) - (A3) are satisfied and $(x^{**}, Y_2(x^{**})) \in S_{12}$, then

$$Y_1(x^{**}) < Y_2(x^{**}) \quad \text{and} \quad 1 < Y_2(x^{**}). \qquad (5.6)$$

*Proof.* Indeed, under the hypotheses of this lemma, with the definition of $S_{12}$ and using (2.9), one gets

$$x_P^* \le x^* < x^{**} \le X_1(y_P^{**}) < x_P^{**}. \qquad (5.7)$$

Hence, in this case, $x^{**} \neq x_P^*$ and $x^{**} \neq x_P^{**}$, and consequently, by Lemma 4.1, $Y_1(x^{**}) < Y_2(x^{**})$. The first relation of (5.6) is therefore proved. To get the second relation of (5.6), we recall that by Lemma 4.1, the function $Y_2$ is strictly increasing on the line segment $[x_P^*, X_1(y_P^{**})]$, and we remark that by(5.7) and (2.10) the following relations hold :

$$x_P^* \le 1 \le x^{**} < X_1(y_P^{**}).$$

Hence, $Y_2(1) \le Y_2(x^{**})$ and moreover,

$$Y_2(1) < Y_2(x^{**}) \quad \text{whenever} \quad 1 < x^{**}. \qquad (5.8)$$

Remark now that, with the definition of $Y_1(1)$ and $Y_2(1)$ and since $P(1, 1) = 1$, we have

$$\text{either} \quad Y_1(1) = 1 \le Y_2(1) \quad \text{or} \quad Y_1(1) \le 1 = Y_2(1). \qquad (5.9)$$

With these relations and using (5.8), one gets

$$1 < Y_2(x^{**}) \quad \text{whenever} \quad 1 < x^{**}.$$

Now, to complete the proof of our lemma it is sufficient to show that

$$1 < Y_2(1) \quad \text{whenever} \quad 1 = x^{**}.$$

Suppose that $x^{**} = 1$. Then from the first relation of (5.6) one gets

$$Y_1(1) < Y_2(1), \qquad (5.10)$$

and by Corollary 4.1 and using (5.7), we obtain

$$\phi(1, Y_1(1)) = 1.$$

Since the function $y \mapsto \phi(1, y)$ is strictly increasing, from the last relation and (5.10) it follows that $\phi_1(1, Y_2(1)) > 1$, and since under our hypotheses, $\phi_1(1, 1) \le 1$, this proves that $Y_2(1) \ne 1$. Hence, using again (5.10), we conclude that

$$Y_2(x^{**}) = Y_2(1) > Y_1(1) = 1.$$

Lemma 5.3 is therefore proved.

□

To investigate the cases (h) and i) the following lemma will be used.

**Lemma 5.4.** Suppose that the condition (A1) is satisfied and let $x \in [x_P^*, x_P^{**}]$, $y \in [y_P^*, y_P^{**}]$ and $(x', y') \in D$ be such that

$$x' \le x \le X_1(y) \quad \text{and} \quad y' \le y \le Y_1(x). \qquad (5.11)$$

Then

$$x' = x = X_1(y) \quad \text{and} \quad y' = y = Y_1(x). \qquad (5.12)$$
Proof. Indeed, suppose that the conditions of our lemma are satisfied and let (5.11) holds. Then according to the definition of the line segments $[x_P^*, x_P^{**}]$ and $[y_P^*, y_P^{**}]$, we have $x_P^* \le x' \le x_P^{**}$ and $y_P^* \le y' \le y_P^{**}$, and by Lemma 5.2, the points $(x, Y_1(x))$ and $(X_1(y), y)$ belong to the set $S_{11}$. Using the definition of $S_{11}$, it follows that

$$x_P^* \le x' \le x < X_1(y_P^*) \quad \text{and} \quad y_P^* \le y' \le y \le Y_1(x_P^*). \qquad (5.13)$$

Since by Lemma 4.1, the function $Y_1$ is decreasing on the line segment $[x_P^*, X_1(y_P^*)]$ and the function $X_1$ is decreasing on the line segment $[y_P^*, Y_1(x_P^*)]$, relations (5.13) imply that $Y_1(x') \ge Y_1(x)$ and $X_1(y') \ge X_1(y)$, and consequently, using (5.11) one gets

$$y' \le y \le Y_1(x) \le Y_1(x') \quad \text{and} \quad x' \le x \le X_1(y) \le X_1(y'). \qquad (5.14)$$

Under the hypotheses of our lemma, $(x', y') \in D$ and by Corollary 5.1 and the definition of the line segments $[X_1(y'), X_2(y')]$ and $[Y_1(x'), Y_2(x')]$, we have

$$X_1(y') \le x' \le X_2(y') \quad \text{and} \quad Y_1(x') \le y' \le Y_2(x').$$

When combined with (5.14) these relations prove (5.12). □

## 5.3 Proof of Proposition 2.2

Now we are ready to complete the proof of Proposition 2.2. Under our hypotheses, by Corollary 5.3 applied with $x = x^{**}$ and $y = y^{**}$, one and only one of the cases (a)-(i) of (5.3) holds, and remark that the case (a) is equivalent to the case (B0).

Remark furthermore that the case (B1) implies (b), and conversely, if the case (b) of (5.3) holds, then by Lemma 5.2 applied with $x = x^{**}$ and $y = y^{**}$, the point $(x^{**}, y^{**})$ belongs to the set $S_{22}$, and by Lemma 4.1,

$$x^{**} < x_P^{**} \quad \text{and} \quad y^{**} < y_P^{**}$$

because in this case, we have $Y_1(x^{**}) < Y_2(x^{**})$ and $X_1(y^{**}) < X_2(y^{**})$. The case (B1) is therefore equivalent to the case (b).

Similarly, the case (B2) implies the case (c) of (5.3) and conversely, in the case (c), by Lemma 5.2 applied with $x = x^{**}$ and $y = y^{**}$, the points $(x^{**}, Y_2(x^{**}))$ and $(X_2(y^{**}), y^{**})$ belong to the set $S_{22}$. The case (B2) is therefore equivalent to the case (c) of (5.3).

And similarly, the case (B3) (resp. (B4)) implies the case (d) (resp (e)), and conversely, if (d) (resp. (e)) holds, then by Lemma 5.2 applied with $x = x^{**}$ and $y = y^{**}$ the point $(x^{**}, y^{**}) = (X_1(y^{**}), y^{**}) = (x^{**}, Y_2(x^{**}))$ belongs to the set $S_{12}$ (resp. the points $(x^{**}, Y_2(x^{**}))$ and $(X_1(y^{**}), y^{**})$ belong to the set $S_{12}$), and by Lemma 5.3 combined with Lemma 4.1 and (2.10), one gets

$$Y_1(x^{**}) < Y_2(x^{**}), \quad x^{**} < x_P^{**} \quad \text{and} \quad y^* \le 1 < Y_2(x^{**}).$$

The case (d) is therefore equivalent to the case (B3), and the case (e) is equivalent to (B4). Similar arguments (it is sufficient to exchange the roles of $x$ and $y$) show that the case (f) is equivalent to (B5) and the case (g) is equivalent to (B6).

Now, to complete the proof of our proposition it is sufficient to show that the case (i) of (5.3) never holds, and the case (h) is equivalent to (B7). To get this result we apply Lemma 5.4 with $(x', y') = (1, 1)$ and $(x, y) = (x^{**}, y^{**})$. By relations (2.9) and (2.10), we have

$$x_P^* \le x^* \le 1 \le x^{**} \le x_P^{**} \quad \text{and} \quad y_P^* \le y^* \le 1 \le y^{**} \le y_P^{**}. \qquad (5.15)$$

Hence, when either (h) or (i) holds, i.e. if

$$X_1(y^{**}) \ge x^{**} \quad \text{and} \quad Y_1(x^{**}) \ge y^{**},$$
using Lemma 5.4 with $x = x^{**}$, $y = y^{**}$ and $(x', y') = (1, 1)$ we obtain

$$ (x^{**}, y^{**}) = (1, 1) \quad \text{and} \quad X_1(1) = Y_1(1) = 1. \qquad (5.16) $$

The case (i) is therefore impossible and the case (h) is equivalent to (5.16). Remark moreover that by Lemma 5.2 applied with $x = x^{**} = 1$ and $y = y^{**} = 1$, from (5.16) it follows that the point $(1, 1)$ belongs to the curve $S_{11}$ and consequently, by Lemma 4.1,

$$ \partial_x P(1, 1) \le 0 \quad \text{and} \quad \partial_y P(1, 1) \le 0, \qquad (5.17) $$

and notice that by (5.16) and using (2.9) and (2.10),

$$ x_P^* \le x^* < x^{**} = 1 \quad \text{and} \quad y_P^* \le y^* < y^{**} = 1. $$

Since the points $(x, y) = (x_P^*, Y_1(x_P^*))$ and $(x', y') = (X_1(y_P^*), y_P^*)$ are the only points in $S_{11}$ for which $\partial_y P(x, y) = 0$ and $\partial_x P(x', y') = 0$, the last relations show that $\partial_x P(1, 1) \ne 0$ and $\partial_y P(1, 1) \ne 0$. Using (5.17) we conclude therefore that when (h) holds, we have also $\partial_x P(1, 1) < 0$ and $\partial_y P(1, 1) < 0$ and consequently the case (h) is equivalent to (B7).

# 6 The Functional Equation and the Convergence Domain

## 6.1 Sketch of the proof of Theorem 2.1

The main ideas of the proof of Theorem 2.1 are the following: By using the method of Lyapunov functions, we first show that the series

$$ H_j(x, y) = \sum_{k=(k_1, k_2) \in \mathbb{Z}_+^2} g(j, k) x^{k_1} y^{k_2}, \quad j \in \mathbb{Z}_+^2 \qquad (6.1) $$

(and consequently, also the series (2.28) and (2.29)) converge on a suitable polycircular set $\Omega(\Theta)$ closed to the points $(x_d, 0)$ and $(0, y_d)$. This is a subject of Proposition 6.1 below. With this preliminary result, we will be able to get the first assertion of Theorem 2.1 and to introduce on the set $\Omega(\Theta)$, the functional equation (2.30) (see Proposition 6.3). Next, we show that the functions at the right hand side of (2.30) are analytic in the set $\{(x, y) \in \Omega(\Gamma) : |x| < x_d, |y| < y_d\}$, and we extend in this way, first the function $(x, y) \mapsto R_j(x, y) = Q(x, u)h_j(x, y)$ and next the function $(x, y) \mapsto h_j(x, y)$ as analytic functions to the set $\{(x, y) \in \Omega(\Gamma) : |x| < x_d, |y| < y_d\}$.

**Definition 8.** If one of the cases (B0)-(B2) holds, we define $\Theta$ as the logarithmically convex hull of the union of the two rectangles $[0, x_d] \times [0, Y_1(x_d)]$ and $[0, X_1(y_d)] \times [0, y_d]$ (see Figure 10) :

$$ \Theta = \text{LogCH} \left\{ ([0, x_d] \times [0, Y_1(x_d)]) \cup ([0, X_1(y_d)] \times [0, y_d]) \right\}. \qquad (6.2) $$

In the case when one of the assertions (B3)-(B6) holds, we let

$$ \Theta = [0, x_d] \times [0, y_d]. \qquad (6.3) $$

**Proposition 6.1.** Under the hypotheses (A1)-(A4), the series (6.1) converges on the set

$$ \Omega(\Theta) = \{(x, y) \in \mathbb{C}^2 : (|x|, |y|) \in \Theta\}. \qquad (6.4) $$

In the case when the random walk $(Z(n))$ is positive recurrent, this statement follows from the results of Miyazawa [26]. In Section 6.2, we give another proof of this statement by using the method of Lyapunov functions. Our proof is valid both for positive recurrent and for transient random walks.

Since for any $j \in \mathbb{Z}_+^2$, and $(x, y) \in ]0, +\infty[^{2}$,

$$ xyh_j(x, y) \le H_j(x, y), \quad xh_{1j}(x) = H_j(x, 0) \le H_j(x, y) \quad \text{and} \quad yh_{2j}(y) = H_j(0, y) \le H_j(x, y), $$

as a straightforward consequence of Proposition 6.1, one gets
**Corollary 6.2.** Under the hypotheses (A1)-(A4), for any $j \in \mathbb{Z}^2$,

i) the series (2.29) converge (and consequently the functions $h_{1j}$ and $h_{2j}$ are analytic) respectively in $B(0, x_d)$ and in $B(0, y_d)$, with $x_d$ and $y_d$ defined respectively by (2.21) and (2.22);

ii) the series (2.28) converge on the set $\Omega(\Theta)$ and consequently, the function $h_j$ is analytic in $\Omega(\Theta)$.

With this results, using classical arguments (see Section 6.3) we obtain

**Proposition 6.3.** Under the hypotheses (A1)-(A4), for any $j \in \mathbb{Z}^2$ and $(x, y) \in \Omega(\Theta)$, the functional equation (2.30) holds.

By the definition of the points $x_d$ and $y_d$, see relations (2.21) and (2.22), and of the sets $\Theta$ and $\Gamma$, when either (B3) or (B4) holds, we have the relations

$$x_d = x^{**}, \quad y_d = Y_2(x_d) \quad \text{and} \quad \Theta = [0, x_d[ \times [0, Y_2(x_d)] = \{(x, y) \in \Gamma : x < x_d, y < y_d\},$$

and when either (B5) or (B6) holds,

$$y_d = y^{**}, \quad x_d = X_2(y_d) \quad \text{and} \quad \Theta = [0, X_2(y_d)] \times [0, x_d[ = \{(x, y) \in \Gamma : x < x_d, y < y_d\}.$$

Hence, when one of the case (B3)-(B6) occurs, Theorem 2.1 follows from Corollary 6.2 and Proposition 6.3.

When one of the cases (B0)-(B2) occurs, the first assertion of Theorem 2.1 follows from Corollary 6.2 as well, and to prove the second assertion of Theorem 2.1 the following proposition will be used.

**Proposition 6.4.** If the conditions (A1)-(A3) are satisfied and one of the assertions (B0)-(B2) holds, then

$$\{(x, y) \in \Gamma : x < x_d, y < y_d\} \subset \Theta \cup \{(x, y) \in \mathring{D} : x < x_d, y < y_d\}. \quad (6.5)$$

With this lemma and using our previous results we will be able to show first that the function $(x, y) \mapsto h_j(x, y)$ can be continued to the set $\Omega_d(\gamma)$ as a meromorphic function, and next to show that it is in fact analytic in this set.

The proof of Theorem 2.1 is organized as follows:

- Proposition 6.1 is proved in Section 6.2;

- Section 6.3 is devoted to the proof of Proposition 6.3;

- the proof of Proposition 6.4 is given in Section 6.4;

- the proof of Theorem 2.1 is completed in Section 6.5.

## 6.2 Proof of Proposition 6.1

### 6.2.1 The main idea of the proof.

To main tool of our proof of this proposition is the method of Lyapunov functions. We show that for any $(x_1, y_1) \in \mathring{D} \cap \mathring{D}_1$ and $(x_2, y_2) \in \mathring{D} \cap \mathring{D}_2$ such that

$$x_1 > x_2 \quad \text{and} \quad y_1 < y_2, \quad (6.6)$$

the function $f: \mathbb{Z}_+^2 \to [0, +\infty[$ defined by

$$f(k_1, k_2) = x_1^{k_1} y_1^{k_2} + x_2^{k_1} y_2^{k_2}, \quad (k_1, k_2) \in \mathbb{Z}_+^2. \quad (6.7)$$
satisfies

$$ \mathbb{E}_j(f(Z(1))) \le \theta f(j), \quad \forall j \in \mathbb{Z}_+^2 \setminus E, \qquad (6.8) $$

with some $0 < \theta < 1$ and for some finite subset $E \subset \mathbb{Z}_+^2$. This is a subject of lemma 6.1 below. With this result we are able to prove that for any two points $(x_1, y_1) \in \mathring{D} \cap \mathring{D}_1$ and $(x_2, y_2) \in \mathring{D} \cap \mathring{D}_2$ satisfying (6.6), the series

$$ \sum_{k=(k_1, k_2) \in \mathbb{Z}_+^2 \setminus \{(0,0)\}} g(j, k)(x_1^{k_1} y_1^{k_2} + x_2^{k_1} y_2^{k_2}) \qquad (6.9) $$

converges, this is a subject of Lemma 6.2. Next, to prove that the series (6.1) converge for any $(x, y) \in \Theta$, we show that for any $(x, y) \in \Theta$, there are two points $(x_1, y_1) \in \mathring{D} \cap \mathring{D}_1$ and $(x_2, y_2) \in \mathring{D} \cap \mathring{D}_2$ satisfying (6.6) and such that

$$ x^{k_1} y^{k_2} \le x_1^{k_1} y_1^{k_2} + x_2^{k_1} y_2^{k_2}, \quad \forall k = (k_1, k_2) \in \mathbb{Z}_+^2. \qquad (6.10) $$

### 6.2.2 Preliminary results for the proof of Proposition 6.1.

We begin our analysis with the following statements.

**Lemma 6.1.** Suppose that the conditions (A1)-(A3) are satisfied and let two points $(x_1, y_1) \in \mathring{D} \cap \mathring{D}_1$ and $(x_2, y_2) \in \mathring{D} \cap \mathring{D}_2$ satisfy (6.6). Then for some finite set $E \subset \mathbb{Z}_+^2$, the function $f: \mathbb{Z}_+^2 \to [0, +\infty[$ defined by (6.7) satisfies (6.8) with some $0 < \theta < 1$.

*Proof.* Consider two functions $f_1, f_2: \mathbb{Z}_+^2 \to \mathbb{R}$ defined by

$$ f_1(j_1, j_2) = x_1^{j_1} y_1^{j_2}, \quad \text{and} \quad f_2(j_1, j_2) = x_2^{j_1} y_2^{j_2}, \quad \forall (j_1, j_2) \in \mathbb{Z}_+^2, $$

and let

$$ \tilde{\theta} = \max\{\phi_1(x_1, y_1), P(x_1, y_1), \phi_2(x_2, y_2), P(x_2, y_2)\}. $$

Then $0 < \tilde{\theta} < 1$ because $(x_1, y_1) \in \mathring{D} \cap \mathring{D}_1$ and $(x_2, y_2) \in \mathring{D} \cap \mathring{D}_2$, and moreover, for any $(j = (j_1, j_2) \in \mathbb{Z}_+^2 \setminus \{0\})$,

$$ \mathbb{E}_{(j_1, j_2)}(f_1(Z(1))) \le \begin{cases} \tilde{\theta} f_1(j_1, j_2), & \text{if } j_1 \ne 0, \\ \phi_2(x_1, y_1) y_1^{j_2}, & \text{if } j_1 = 0, \end{cases} $$

and

$$ \mathbb{E}_{(j_1, j_2)}(f_2(Z(1))) \le \begin{cases} \tilde{\theta} f_2(j_1, j_2), & \text{if } j_2 \ne 0, \\ \phi_1(x_2, y_2) x_2^{j_1}, & \text{if } j_2 = 0. \end{cases} $$

For the function $f = f_1 + f_2$, we obtain therefore that for any $j = (j_1, j_2) \in \mathbb{Z}_+^2 \setminus \{0\}$,

$$ \mathbb{E}_{(j_1, j_2)}(f(Z(1))) \le \begin{cases} \tilde{\theta} f(j_1, j_2) & \text{if } j_1 > 0 \text{ and } j_2 > 0 \\ \tilde{\theta} y_2^{j_2} + \phi_2(x_1, y_1) y_1^{j_2} & \text{if } j_1 = 0 \\ \tilde{\theta} x_1^{j_1} + \phi_1(x_2, y_2) x_2^{j_1} & \text{if } j_2 = 0. \end{cases} \qquad (6.11) $$

Since under hypotheses of our lemma, $x_1 > x_2 > 0$ and $0 < y_1 < y_2$, we have moreover

$$ \lim_{j_1} (x_2/x_1)^{j_1} = \lim_{j_2 \to \infty} (y_1/y_2)^{j_2} = 0, $$
and consequently, for any $\varepsilon > 0$ there is $N_\varepsilon > 0$ such that for any $(j_1, j_2) \in \mathbb{Z}_+^2$ with $j_2 > N_\varepsilon$ and $j_1 = 0$,

$$ \tilde{\theta} y_2^{j_2} + \phi_2(x_1, y_1) y_1^{j_2} \le (\tilde{\theta} + \varepsilon) y_2^{j_2} = (\tilde{\theta} + \varepsilon) f_2(j_1, j_2) \le (\tilde{\theta} + \varepsilon) f(j_1, j_2) $$

and similarly, for any $(j_1, j_2) \in \mathbb{Z}_+^2$ with $j_1 > N_\varepsilon$ and $j_2 = 0$,

$$ \tilde{\theta} x_1^{j_1} + \phi_1(x_2, y_2) x_2^{j_1} \le (\tilde{\theta} + \varepsilon) f(j_1, j_2). $$

Hence, for $\varepsilon > 0$ such that $\tilde{\theta} + \varepsilon < 1$, letting

$$ E = \{(j_1, 0) \in \mathbb{Z}_+^2 : j_1 \le N_\varepsilon\} \cup \{(0, j_2) \in \mathbb{Z}_+^2 : j_2 \le N_\varepsilon\} $$

one gets (6.8) with $\theta = \tilde{\theta} + \varepsilon < 1$. $\square$

**Lemma 6.2.** Under the hypotheses of Lemma 6.1, for any $j \in \mathbb{Z}_+^2$, the series (6.9) converges.

*Proof.* Indeed, consider two points $(x_1, y_1) \in \mathring{D} \cap \mathring{D}_1$ and $(x_2, y_2) \in \mathring{D} \cap \mathring{D}_2$ satisfying (6.6) and let the function $f: \mathbb{Z}_+^2 \to [0, +\infty[$ be defined by (6.7). Then by Lemma 6.1, for some finite set $E \subset \mathbb{Z}_+^2$ and some positive number $\theta < 1$, (6.8) holds. Without any restriction of generality, we will suppose that $E$ contains the origin $(0, 0)$. Denote by $\tau_E$ the first time when the process $(Z(n))$ hits the set $E$:

$$ \tau_E = \inf\{n \ge 1 : Z(n) \in E\}, $$

and let

$$ g_E(j, k) = \sum_{n=0}^{\infty} \mathbb{P}_j(Z(n) = k, \tau_E \ge n), \quad j \in \mathbb{Z}_+^2 \setminus E, k \in \mathbb{Z}_+^2. $$

Then by (6.8), for any $j \in \mathbb{Z}_+^2 \setminus E$,

$$ \mathbb{E}_j(f(Z(n)); \tau_E \ge n) \le \theta^n f(j), \quad \forall n \in \mathbb{N}, $$

and consequently,

$$ \sum_{k \in \mathbb{Z}_+^2} g_E(j, k) (x_1^{k_1} y_1^{k_2} + x_2^{k_1} y_2^{k_2}) = \sum_{k \in \mathbb{Z}_+^2} g_E(j, k) f(k) = \sum_{n=1}^{\infty} \mathbb{E}_j(f(Z(n)); \tau_E \ge n) \le f(j) (1 - \theta)^{-1}. \quad (6.12) $$

Using the identity

$$ g(j, k) = g_E(j, k) + \sum_{\ell \in E \setminus \{0\}} \sum_{m \in \mathbb{Z}_+^2 \setminus E} g(j, \ell) \mathbb{P}_\ell(Z(1) = m) g_E(m, k) $$

from the above relation it follows that for any $j \in \mathbb{Z}_+^2 \setminus E$,

$$ \sum_{k=(k_1, k_2) \in \mathbb{Z}_+^2} g(j, k) (x_1^{k_1} y_1^{k_2} + x_2^{k_1} y_2^{k_2}) \le \frac{1}{1-\theta} \left( f(j) + \sum_{\ell \in E \setminus \{0\}} \sum_{m \in \mathbb{Z}_+^2 \setminus E} g(j, \ell) \mathbb{P}_\ell(Z(1) = m) f(m) \right) \\ \le \frac{1}{1-\theta} \left( f(j) + \sum_{\ell \in E \setminus \{0\}} g(j, \ell) \mathbb{E}_\ell(f(Z(1))) \right). $$

Since the set $E$ is finite and under the hypotheses (A1) and (A3), $\mathbb{E}_\ell(f(Z(1))) < +\infty$ for any $\ell \in E$, this proves that the series (6.9) converges for any $j \in \mathbb{Z}_+^2 \setminus E$. To prove that this series converges for $j \in E$, it is sufficient now to notice that for $j \in E$,

$$ g(j, k) = \sum_{\ell \in E \setminus \{0\}} g(j, \ell) \sum_{m \in \mathbb{Z}_+^2 \setminus E} \mathbb{P}_\ell(Z(1) = m) g_E(m, k), $$
and consequently,

$$ \sum_{k=(k_1,k_2) \in \mathbb{Z}_+^2} g(j,k) (x_1^{k_1} y_1^{k_2} + x_2^{k_1} y_2^{k_2}) \\ \le \sum_{\ell \in E \setminus \{0\}} g(j, \ell) \sum_{m \in \mathbb{Z}_+^2 \setminus E} \mathbb{P}_\ell(Z(1) = m) \sum_{k=(k_1,k_2) \in \mathbb{Z}_+^2} g_E(m,k) (x_1^{k_1} y_1^{k_2} + x_2^{k_1} y_2^{k_2}) < +\infty. $$

□

**Lemma 6.3.** Suppose that the conditions (A1)-(A3) are satisfied and let $\tilde{x} \in ]x^*, x^{**}[$ and $\tilde{y} \in ]y^*, y^{**}[$ be such that

$$ \tilde{x} > X_1(\tilde{y}) \quad \text{and} \quad \tilde{y} > Y_1(\tilde{x}). \qquad (6.13) $$

Then for any $y_1 > Y_1(\tilde{x})$ closed enough to $Y_1(\tilde{x})$ and any $x_2 > X_1(\tilde{y})$ closed enough to $X_1(\tilde{y})$, the points $(x_1, y_1) = (\tilde{x}, y_1)$ and $(x_2, y_2) = (x_2, \tilde{y})$ satisfy the conditions of Lemma 6.1.

*Proof.* Indeed, by Lemma 4.2, for $\tilde{x} \in ]x^*, x^{**}[$ and $\tilde{y} \in ]y^*, y^{**}[$, each of the line segments $[Y_1(\tilde{x}), \tilde{Y}_2(\tilde{x})] = \{y > 0 : (\tilde{x}, y) \in D \cap D_1\}$ and $[X_1(\tilde{y}), \tilde{X}_2(\tilde{y})] = \{x > 0 : (x, \tilde{y}) \in D \cap D_2\}$ has a non-zero length, and for any $y_1$ and $x_2$ such that $Y_1(\tilde{x}) < y_1 < \tilde{Y}_2(\tilde{x})$ and $X_1(\tilde{y}) < x_2 < \tilde{X}_2(\tilde{y})$, one has

$$ (\tilde{x}, y_1) \in \mathring{D} \cap \mathring{D}_1, \quad \text{and} \quad (x_2, \tilde{y}) \in \mathring{D} \cap \mathring{D}_2. $$

Using (6.13), this proves that for any $y_1$ and $x_2$ such that $Y_1(\tilde{x}) < y_1 < \min\{\tilde{Y}_2(\tilde{x}), \tilde{y}\}$ and $X_1(\tilde{y}) < x_2 < \min\{\tilde{x}, \tilde{X}_2(\tilde{y})\}$, the points $(x_1, y_1) = (\tilde{x}, y_1)$ and $(x_2, y_2) = (x_2, \tilde{y})$ satisfy the conditions of Lemma 6.1. □

Now we are ready to get

**Lemma 6.4.** Suppose that the conditions (A1)-(A4) are satisfied and let

$$ \Theta_0 = \begin{cases} ([0, x_d[ \times [0, Y_1(x_d)] \cup ([0, X_1(y_d)] \times [0, y_d[) & \text{if either (B0), or (B1) or (B2) holds} \\ [0, x_d[ \times [0, Y_2(x_d)] & \text{if either (B3) or (B4) holds} \\ [0, X_1(y_d)] \times [0, y_d[ & \text{if either (B5) or (B6) holds.} \end{cases} \qquad (6.14) $$

Then for any $(x, y) \in \Theta_0$, there are two points $(x_1, y_1)$ and $(x_2, y_2)$ satisfying the conditions of Lemma 6.1 and relations (6.10).

*Proof.* Suppose first that one of the cases (B0), (B1) or (B2) occurs and let $(x, y) \in [0, x_d[ \times [0, Y_1(x_d)]$. The definitions of the cases (B0), (B1), (B2) and of the points $x_d$ and $y_d$ (see Proposition 2.2 and relations (2.33), (2.34)), give

$$ \max\{X_1(y_d), x\} < x_d = x^{**} \quad \text{and} \quad y < Y_1(x_d) < y_d = y^{**}, $$

and by (2.9),

$$ x_P^* \le x^* < x^{**} = x_d \quad \text{and} \quad y_P^* \le y^* < y^{**} = y_d. $$

Since the functions $X_1 : [y_P^*, y_P^{**}] \to [x_P^*, x_P^{**}]$ and $Y_1 : [x_P^*, x_P^{**}] \to [y_P^*, y_P^{**}]$ are continuous, it follows that for any $\tilde{x} \in ]x^*, x^{**}]$, $x_d[ \text{ and } \tilde{y} \in ]y^*, y^{**}]$ closed enough respectively to $x_d$ and $y_d$ one has

$$ 0 \le x < \tilde{x}, \quad 0 \le y < Y_1(\tilde{x}), \qquad (6.15) $$

$$ X_1(\tilde{y}) < \tilde{x} \quad \text{and} \quad Y_1(\tilde{x}) < \tilde{y}. \qquad (6.16) $$
By Lemma 6.3, from (6.16) it follows that for $y_1 > Y_1(\tilde{x})$ and $x_2 > X_1(\tilde{y})$ closed enough respectively to $Y_1(\tilde{x})$ and $X_1(\tilde{y})$, the points $(x_1, y_1) = (\tilde{x}, y_1)$ and $(x_2, y_2) = (x_2, \tilde{y})$ satisfy the conditions of Lemma 6.1, and using (6.15) one gets (6.10).

When one of the cases (B0), (B1) or (B2) holds and $(x, y) \in [0, x_d[ \times [0, Y_1(x_d)],$ our lemma is therefore proved. For $(x, y) \in [0, X_1(y_d)] \times [0, y_d[$, the proof is quite similar.

Suppose now that either (B3) or (B4) holds. The definitions of the cases (B3) and (B4) and of the points $x_d$ and $y_d$ (see Proposition 2.2 and relations (2.33), (2.34)), and by (2.9), give

$$x_P^* \le x^* < x^{**} = x_d \le X_1(y^{**}), \qquad (6.17)$$

$$y^* \le 1 < y_d = Y_2(x_d) \le y^{**}, \qquad (6.18)$$

$$(X_1(y^{**}), y^{**}), (x^{**}, Y_2(x^{**})) = (x_d, y_d) \in S_{12}, \qquad (6.19)$$

and

$$Y_1(x_d) < Y_2(x_d). \qquad (6.20)$$

From the definition of $S_{12}$ and relation (6.19), it follows that $X_1(y^{**}) \le X_1(y_P^{**})$ and consequently, by (6.17),

$$x_P^* \le x^* < x_d \le X_1(y^{**}) \le X_1(y_P^{**}). \qquad (6.21)$$

Consider now a point $(x, y) \in \Theta_0$. Then by (6.14) and using (6.21), (6.18) and (6.20),

$$0 \le \max\{x^*, x\} < x_d \le X_1(y_P^{**}) \quad \text{and} \quad 0 \le \max\{Y_1(x_d), y^*, y\} < y_d = Y_2(x_d) \le y^{**} \qquad (6.22)$$

Since the function $Y_2$ is strictly increasing on the line segment $[x_P^*, X_1(y_P^{**})]$ and the functions $Y_1$ and $Y_2$ are continuous on $[x_P^*, x_P^{**}]$, it follows that for any $\hat{x}$ and $\tilde{x}$ closed enough to $x_d$ and such that

$$\max\{x^*, x\} < \hat{x} < \tilde{x} < x^{**} = x_d, \qquad (6.23)$$

one has

$$\max\{Y_1(\tilde{x}), y^*, y\} < Y_2(\hat{x}) < Y_2(\tilde{x}) < Y_2(x_d) = y_d \le y^{**},$$

Remark that because of (6.21) and (6.23), the points $\hat{x}$ and $\tilde{x}$ belong to the line segment $[x_P^*, X_1(y_P^{**})]$. Since the functions $X_1: [Y_1(x_P^*), y_P^{**}] \to [x_P^*, X_1(y_P^{**})]$ and $Y_2: [x_P^*, X_1(y_P^{**})] \to [Y_1(x_P^*), y_P^{**}]$ are inverse to each other, letting $\tilde{y} = Y_2(\hat{x})$ we get therefore $X_1(\tilde{y}) = \hat{x}$ and using the above relations we obtain

$$y^* < \tilde{y} = Y_2(\hat{x}) < y^{**}, \quad Y_1(\tilde{x}) < Y_2(\hat{x}) = \tilde{y}, \quad X_1(\tilde{y}) = \hat{x} < \tilde{x}, \qquad (6.24)$$

$$x < \hat{x} = X_1(\tilde{y}) \quad \text{and} \quad y < Y_2(\hat{x}) = \tilde{y}. \qquad (6.25)$$

By Lemma 6.3, from (6.23) and (6.24) it follows that for any $y_1 > Y_1(\tilde{x})$ closed enough to $Y_1(\tilde{x})$ and $x_2 > X_1(\tilde{y})$ closed enough to $X_1(\tilde{y})$, the points $(x_1, y_1) = (\tilde{x}, y_1)$ and $(x_2, y_2) = (x_2, \tilde{y})$ satisfy the conditions of Lemma 6.1, and using moreover (6.25), we get that for any $j = (j_1, j_2) \in \mathbb{Z}_+^2$,

$$x^{j_1} y^{j_2} \le (X_1(\tilde{y}))^{j_1} \tilde{y}^{j_2} \le x_2^{j_1} y_2^{j_2} \le x_1^{j_1} y_1^{j_2} + x_2^{j_1} y_2^{j_2}.$$

Hence, in the case when either (B3) or (B4) holds, Lemma 6.3 is also proved. To prove this lemma the case when either (B5) or (B6) holds, it is sufficient to exchange the roles of $x$ and $y$. $\square$
### 6.2.3 Proof of Proposition 6.1

This proposition is a consequence of Lemma 6.2 and Lemma 6.3: by Lemma 6.3, for any $(x, y) \in \Theta_0$, there are two points $(x_1, y_1)$ and $(x_2, y_2)$ for which the conditions of Lemma 6.1 are satisfied and relations (6.10) hold. By Lemma Lemma 6.2 and using (6.10), this proves that

$$ \sum_{k=(k_1, k_2) \in \mathbb{Z}_+^2} g(j, k) x^{k_1} y^{k_2} \le \sum_{k=(k_1, k_2) \in \mathbb{Z}_+^2} g(j, k) (x_1^{k_1} y_1^{k_2} + x_2^{k_1} y_2^{k_2}) < +\infty. $$

Hence, for any $(x, y) \in \Theta_0$, the series (6.1) converge. Since the set $\Theta$ is the logarithmically convex hull of the set $\Theta_0$, and the domain of convergence of power series with center 0 is always logarithmically convex, this proves that the series (6.1) converge in $\Omega(\Theta)$.

## 6.3 Proof of Proposition 6.3

Consider first the case when $j = (j_1, j_2) \ne (0, 0)$. By Proposition 6.1, the series (6.1) converge on the set $\Omega(\Theta)$. Hence, for any $(x, y) \in \Omega(\Theta)$ with non-zero $x$ and $y$, by the Fubini theorem and using the Markov property, one gets

$$ \begin{aligned} H_j(x, y) &= x^{j_1} y^{j_2} + \sum_{k=(k_1, k_2) \in \mathbb{Z}_+^2} \sum_{n=1}^{\infty} \mathbb{P}_j(Z(n) = k, \tau_0 > n) x^{k_1} y^{k_2} \\ &= x^{j_1} y^{j_2} + \sum_{\ell \in \mathbb{Z}_+^2 \setminus \{(0,0)\}} g(j, \ell) \mathbb{E}_{\ell} \left( x^{Z_1(1)} y^{Z_2(1)}, \tau_0 > 1 \right) \end{aligned} \quad (6.26) $$

Because of Assumptions (A1)-(A3), for $(x, y) \in \Omega(\Theta)$ with $x \ne 0$ and $y \ne 0$, we have

$$ \mathbb{E}_{\ell} \left( x^{Z_1(1)} y^{Z_2(1)}, \tau_0 > 1 \right) = \begin{cases} x^{\ell_1} y^{\ell_2} P(x, y) - \mu(-1, -1) \mathbb{1}_{\{(1,1)\}}(\ell_1, \ell_2) & \text{if } \ell_1 > 0 \text{ and } \ell_2 > 0, \\ x^{\ell_1} \phi_1(x, y) - \mu_1(-1, 0) \mathbb{1}_{\{(1,0)\}}(\ell_1, \ell_2) & \text{if } \ell_1 > 0 \text{ and } \ell_2 = 0, \\ y^{\ell_2} \phi_2(x, y) - \mu_2(0, -1) \mathbb{1}_{\{(0,1)\}}(\ell_1, \ell_2) & \text{if } \ell_1 = 0 \text{ and } \ell_2 > 0. \end{cases} $$

Using these relations in (6.26), for $(x, y) \in \Omega(\Theta)$ with non-zero $x$ and $y$, one gets therefore

$$ H_j(x, y) = x^{j_1} y^{j_2} - \mu(-1, -1)g(j, (1, 1)) - \mu_1(-1, 0)g(j, (1, 0)) - \mu_2(0, -1)g(j, (0, 1)) \\ + P(x, y) \sum_{\substack{\ell=(\ell_1, \ell_2) \in \mathbb{Z}_+^2: \\ \ell_1 > 0, \ell_2 > 0}} g(j, \ell) x^{\ell_1} y^{\ell_2} + \phi_1(x, y) \sum_{\ell_1=1}^{\infty} g(j, (\ell_1, 0)) x^{\ell_1} + \phi_2(x, y) \sum_{\ell_2=1}^{\infty} g(j, (0, \ell_2)) y^{\ell_2} $$

or equivalently,

$$ H_j(x, y) = L_j(x, y) + xyP(x, y)h_j(x, y) + x\phi_1(x, y)h_{1j}(x) + y\phi_2(x, y)h_{2j}(y) $$

with

$$ \begin{aligned} L_j(x, y) &= x^{j_1} y^{j_2} - \mu(-1, -1)g(j, (1, 1)) - \mu_1(-1, 0)g(j, (1, 0)) - \mu_2(0, -1)g(j, (0, 1)) \\ &= x^{j_1} y^{j_2} - \mathbb{P}_j(\tau_0 < +\infty). \end{aligned} $$

Since clearly $H_j(x, y) = xyh_j(x, y) + xh_{1j}(x) + yh_{2j}(y)$, the last relation proves (2.30) for any $(x, y) \in \Omega(\Theta)$ and $j = (j_1, j_2) \in \mathbb{Z}_+^2 \setminus \{(0, 0)\}$. To get (2.30) for $(j_1, j_2) = (0, 0)$, it is sufficient now to notice that for any $(x, y) \in \Omega(\Theta)$,

$$ H_{(0,0)}(x, y) = \sum_{(j_1, j_2) \in \mathbb{Z}_+^2 \setminus \{(0,0)\}} \mathbb{P}_{(0,0)}(Z(1) = (j_1, j_2)) H_j(x, y) $$
and

$$ \sum_{(j_1, j_2) \in \mathbb{Z}_+^2 \setminus \{(0,0)\}} \mathbb{P}_{(0,0)}(Z(1) = (j_1, j_2)) L_j(x, y) = \mathbb{E}_{(0,0)}(x^{Z_1(1)} y^{Z_2(1)}; \tau_0 > 1) - \mathbb{P}_{(0,0)}(1 < \tau_0 < +\infty) \\ = \phi_0(x, y) - \mathbb{P}_{(0,0)}(\tau_0 < +\infty). $$

## 6.4 Proof of Proposition 6.4

Suppose that one of the cases (B0)-(B2) holds. Then by Proposition 2.2 and the definition of the points $x_d, y_d$, the following relations hold

$$ x_d = x^{**} > X_1(y_d), \quad y_d = y^{**} > Y_1(x_d), \qquad (6.27) $$

and the set $\Theta$ is defined as a logarithmically convex hull of the set

$$ ([0, x_d[ \times [0, Y_1(x_d)]] \cup ([0, X_1(y_d)] \times [0, y_d]). $$

The points $(x_1, y_2) = (x_d, Y_1(x_d))$ and $(x_2, y_2) = (X_1(y_d), y_d)$ are therefore on the boundary of the set $\Theta$ and also on the boundary of the set $D$. Since the set $\Theta$ is logarithmically convex and the set $D$ is strictly logarithmically convex, it follows that for any $0 < \theta < 1$, the point $(x_\theta, y_\theta)$, with $x_\theta = x_1^\theta x_2^{1-\theta}$ and $y_\theta = y_1^\theta y_2^{1-\theta}$, belongs to the set $\bar{\Theta} \cap \mathring{D}$, and consequently,

$$ Y_1(x_\theta) < y_\theta < Y_2(x_\theta). $$

By the definition of the set $\Theta$, for any $0 < \theta < 1$ and $y \in [0, y_\theta]$, the point $(x_\theta, y)$ is in $\Theta$, and for any $y \in [Y_1(x_\theta), Y_2(x_\theta)]$ and the point $(x_\theta, y)$ is in $\mathring{D}$. Consequently, for any $0 < \theta < 1$ and $y \in [0, Y_2(x_\theta)]$, the point $(x_\theta, y)$ belongs to the set $\bar{\Theta} \cup \mathring{D}$, or equivalently, that for any $y \ge 0$,

$$ y \ge Y_2(x_\theta) \quad \text{whenever} \quad (x_\theta, y) \notin \bar{\Theta} \cup \mathring{D}. \qquad (6.28) $$

Consider now a point $(\hat{x}, \hat{y}) \in [0, +\infty]^2$ with $\hat{x} < x_d$ and $\hat{y} < y_d$ and such that $(\hat{x}, \hat{y}) \notin (\Theta \cup \mathring{D})$. Then $\hat{x} \ge X_1(y_d)$ (because otherwise $(\hat{x}, \hat{y}) \in [0, X_1(y_d)] \times [0, y_d] \subset \Theta$), and consequently for some $0 < \theta \le 1$, $\hat{x} = x_\theta = x_d^\theta (X_1(y_d))^{1-\theta}$. Hence, by (6.28), we get $\hat{y} \ge Y_2(\hat{x})$, and with similar arguments (it is sufficient to exchange the roles of $x$ and $y$) we obtain $\hat{x} \ge X_2(\hat{y})$. By Lemma 5.2, these inequalities show that $(\hat{x}, Y_2(\hat{x}))$, $(X_2(\hat{y}), \hat{y}) \in S_{22}$. Hence, if we suppose that for some $(x', y') \in D$, the inequalities $\hat{x} \le x'$ and $\hat{y} \le y'$ hold, then using the similar arguments as in the proof of Lemma 5.4, we will obtain that $\hat{x} = x' = X_2(\hat{y})$ and $\hat{y} = y' = Y_2(\hat{x})$. With the definition of the set $\Gamma$, this proves that $(\hat{x}, \hat{y}) \notin \Gamma$.

## 6.5 Proof of Theorem 2.1

Now, we are ready to complete the proof of Theorem 2.1. In the cases (B3)-(B6), Theorem 2.1 follows from Corollary 6.2 and Proposition 6.3, and in the cases (B0)-(B2), the first assertion of Theorem 2.1 follows from Corollary 6.2 as well. To complete the proof of Theorem 2.1 we have therefore to prove its second assertion in the cases (B0)-(B2).

We know that, for any $j \in \mathbb{Z}_+^2$, the function $(x, y) \mapsto L_j(x, y)$ is analytic in $\Omega(\Gamma)$, the functions $(x, y) \mapsto Q(x, y) = xy(1 - P(x, y))$, $(x, y) \mapsto \psi_1(x, y) = x(\phi_1(x, y) - 1)$ and $(x, y) \mapsto \psi_2(x, y) = y(1 - \phi_2(x, y))$ can be analytically continued to the set $\Omega(\Gamma)$, and by Corollary 6.2, the functions $h_{1j}$ and $h_{2j}$ are analytic respectively in the discs $B(0, x_d)$ and $B(0, y_d)$. Hence for any $j \in \mathbb{Z}_+^2$, the function $(x, y) \mapsto L_j(x, y) + \psi_1(x, y)h_{1j}(x) + \psi_2(x, y)h_{2j}(y)$ is analytic in the set $\{(x, y) \in \Omega(\Gamma) : |x| < x_d, |y| < y_d\}$. Since by Proposition 6.3, on the set $\Omega(\Theta)$,

$$ Q(x, y)h_j(x, y) = L_j(x, y) + \psi_1(x, y)h_{1j}(x) + \psi_2(x, y)h_{2j}(y), $$
and since clearly $\Theta \subset \{(x, y) \in \Gamma : x < x_d, y < y_d\}$, we conclude therefore that the functions $(x, y) \mapsto Q(x, y)h_j(x, y)$ can be analytically continued to the set

$$ \Omega_d(\Gamma) \stackrel{\text{def.}}{=} \{(x, y) \in \Omega(\Gamma) : |x| < x_d, |y| < y_d\} $$

and the function $(x, y) \mapsto h_j(x, y)$ can be continued as a meromorphic function

$$ h_j(x, y) = \frac{L_j(x, y) + \psi_1(x, y)h_{1j}(x) + \psi_2(x, y)h_{2j}(y)}{Q(x, y)} $$

to the set $\Omega_d(\Gamma)$. Since

- by Proposition 6.1, for any $j \in \mathbb{Z}_+^2$, the function $h_j$ is analytic in $\Omega(\Theta)$;

- for any $(x, y) \in \Omega(\mathring{D})$, by the definition of the set $D$, we have $|P(x, y)| \le P(|x|, |y|) < 1$ and, consequently, $Q(x, y) \ne 0$;

- and by Proposition 6.4, the set $\Omega_d(\Gamma)$ is included to the union of the open sets $\Omega(\Theta)$ and $\Omega(\mathring{D})$,

we conclude therefore that the function $h_j$ can be analytically continued to the set $\Omega_d(\Gamma)$.

# 7 Singularity Analysis of Generating Functions

This section is devoted to the proof of Theorem 2.3. Recall that the case (B7) will not be considered. See Section 2.2. Throughout this section we will assume therefore that

**Assumption (A4)** one of the cases (B0)-(B6) holds.

## 7.1 The main ideas and the sketch of the proof

The main steps of our proof are the following. First, we prove the light version of the assertion (i)-(v) of this theorem, i.e. we get, in each of the corresponding cases, relations (2.40), (2.44) (2.42), (2.46), (2.48), (2.50), (2.52), and (2.54) with positive constants given by (2.41), (2.43), (2.45), (2.47), (2.49), (2.51), (2.53). This is a subject of Propositions 7.10 and 7.12 and Lemma 7.6 below.

With these results we will be able to show that in each of the assertions (i)-(vi) of our theorem, the corresponding function $\kappa_i, \tilde{\kappa}_i$ or $\kappa_{(\hat{x}, \hat{y})}$ is non-negative on $\mathbb{Z}_+^2$.

Second, in Proposition 7.13 below, we prove that in each of the assertions (i)-(vi), the corresponding function $\kappa_i, \tilde{\kappa}_i$ or $\kappa_{(\hat{x}, \hat{y})}$ is harmonic for the random walk $(Z_{\tau_0}(n))$ and positive everywhere in the set $\mathbb{Z}_+^2 \setminus E_0$. With this statement, the proof of our theorem will be completed.

The first step of the proof of the last assertion of Theorem 2.3 is given by Lemma 7.6. This result is obtained as a straightforward consequence of Theorem 2.1.

The first step of the proof of the assertions (i)-(v) of Theorem 2.3 proof is the most difficult. Its main idea is the following: we extend the function $Y_1$ as an analytic function to some domain of $\mathbb{C}$ large enough and we inject next $y = Y_1(x)$ to the functional equation (2.30) in order to get the identity

$$ L_j(x, Y_1(x)) + \psi_1(x, Y_1(x))h_{1j}(x) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x)) = 0. \quad (7.1) $$

Using this result we will be able to extend beyond the point $x_d$ first the function $x \mapsto \psi_1(x, Y_1(x))h_{1j}(x) = (\phi_1(x, Y_1(x) - 1)H_j(x, 0)$ and next the function $x \mapsto H_j(x, 0)$.

Remark that in a difference with nearest neighbor random walks, we have no explicit form of the function $Y_1$. Moreover, by the implicit function theorem, this function can be extended
only to some neighborhood of the interval $]x_P^*, x_P^{**}]$ in $\mathbb{C}$ which is clearly not sufficient for our purpose. The first difficulty of our proof is therefore to extend the function $Y_1$ to some sufficient for our analysis domain of $\mathbb{C}$. We perform this first step of our proof by using the probabilistic representation of the function $Y_1: [x_P^*, x_P^{**}] \to [y_P^*, y_P^{**}]$ obtained in [16]. This is a subject of Proposition 7.1 below. In this statement, for some $\varepsilon > 0$, we extend the function $Y_1$ as an analytic function to the set

$$ \mathring{U}_{\varepsilon} = C(x_P^*, x_P^{**} + \varepsilon)\setminus[x_P^{**}, x_P^{**} + \varepsilon[, $$

and a continuous function on the set

$$ U_{\varepsilon} = C(x_P^*, x_P^{**} + \varepsilon)\setminus[x_P^{**}, x_P^{**} + \varepsilon[ = \mathring{U}_{\varepsilon} \cup \{x_P^{**}\}. \quad (7.2) $$

It is proved moreover that the extended function $Y_1$ satisfies the identity

$$ Q(x, Y_1(x)) = 0, \quad (7.3) $$

on the set $U_{\varepsilon}$, and that, on the closed annulus $\bar{C}(x_P^*, x_P^{**})$, the following relations holds

$$ |Y_1(x)| < Y_1(|x|) \quad \text{whenever } x \neq |x|. \quad (7.4) $$

Another difficulty is that by Theorem 2.1, we know only that the function $y \mapsto h_{2j}(y)$ is analytic in $B(0, y_d)$, and the function $(x, y) \mapsto h_j(x, y)$ is analytic in $\{(x, y) \in \Omega(\Gamma) : |x| < x_d, |y| < y_d\}$. Hence, we can inject $y = Y_1(x)$ to the functional equation (2.30) only for those $x \in B(0, x_d)$ for which $|Y_1(x)| < y_d$. To overcome this difficulty, we use the inequality (7.4). With this inequality, we are able to show that for some $\delta > 0$, the function $x \mapsto (x, Y_1(x))$ maps the annulus $C(x_d - \delta, x_d) = \{x \in \mathbb{C} : x_d - \delta < |x| < x_d\}$ to the set $\{(x, y) \in \Omega(\Gamma) : |x| < x_D, |y| < y_D\}$ where the functions $(x, y) \mapsto H_j(x, y)$, $(x, y) \mapsto L_j(x, y) + \psi_2(x, y)h_{2j}(y)$ and $(x, y) \mapsto \psi_1(x, y)h_{1j}(x)$ are analytic. This is a subject of Proposition 7.4 below.

In this way, on the annulus $C(x_d - \delta, x_d)$, by letting in the functional equation (2.30) $y = Y_1(x)$ and using (7.3) we obtain the identity (7.1) with analytic in $C(x_d - \delta, x_d)$ functions $x \mapsto \psi_1(x, Y_1(x))h_{1j}(x)$ and $x \mapsto L_j(x, Y_1(x))$ and $x \mapsto \psi_2(x, Y_1(x))h_{2j}(Y_1(x))$. Moreover, when one of the cases (B0)-(B4) holds, using again the inequality (7.4), we will be able to show that for some $\delta > 0$, the function $(x, y) \mapsto L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x))$ is analytic in $C(x_d - \delta, x_d + \delta) \cap \mathring{U}_{\varepsilon}$ and continuous on $C(x_d - \delta, x_d + \delta) \cap U_{\varepsilon}$. Using this result together with the identity (7.1), we will extend the function $x \mapsto -\psi_1(x, Y_1(x))h_{1j}(x) = (1 - \phi_1(x, Y_1(x))H_j(x, 0)$ as an analytic function to the set $C(x_d - \delta, x_d + \delta) \cap \mathring{U}_{\varepsilon}$ and a continuous function to the set $C(x_d - \delta, x_d + \delta) \cap U_{\varepsilon}$. This is a subject of Corollary 7.6 below.

Next, in Proposition 7.7, we investigate the function $x \mapsto 1 - \phi_1(x, Y_1(x)))$, and finally, in Proposition 7.10 and Proposition 7.11 below, when one of the cases (B0)-(B4) holds, we extend the function $x \mapsto H_j(x, 0)$ beyond the point $x_d$.

In the case when either (B5) or (B6) holds, we begin our analysis by investigating the function $h_{2j}$. With the same arguments as in the previous cases (it is sufficient to exchange the roles of $x$ and $y$) we obtain that the function $h_{2j}$ can be extended as an analytic function to the set $B(0, y_d + \delta_0)\setminus\{y_d\}$. Next we show that for some $\delta > 0$, the function $x \mapsto h_{2j}(Y_1(x))$ is analytic in the set $C(x_d - \delta, x_d + \delta) \cap U_{\varepsilon}\setminus\{x_d\}$. And finally, using again the identity (7.1) we extend first the function $x \mapsto \psi_1(x, Y_1(x))h_{1j}(x)$ and next the function $x \mapsto h_{1j}(x)$ beyond the point $x_d$. This is a subject of Proposition 7.12 below.
## 7.2 Analytic continuation and properties of the function $x \mapsto Y_1(x)$

We begin our analysis with the following result.

**Proposition 7.1. Analytic continuation of the function $Y_1$.** Suppose the conditions (A1) are satisfied and let $\mu(j) = 0$ for all $j \in \mathbb{Z}^2$ with $j_2 < -1$ (remark that we do not need the whole condition (A2) to be satisfied). Then

i) the function $Y_1$ is strictly convex on the line segment $[x_P^*, x_P^{**}]$ and for some $\varepsilon > 0$, it can be extended to the set $U_\varepsilon$ as an analytic function in the set $\tilde{U}_\varepsilon$ and a continuous function on the set $U_\varepsilon$ satisfying there the identity (7.3).

ii) relation (7.4) holds on the set $\bar{C}(x_P^*, x_P^{**})$;

iii) for any $\hat{x} \in ]X_1(y_P^*), x_P^{**}[$, the function $x \mapsto (Y_1(x) - Y_1(\hat{x}))$ does not vanish in the set $C(\hat{x} - \delta, \hat{x} + \delta)\setminus\{\hat{x}\}$ for some $\delta > 0$, and has at the point $\hat{x}$ a simple zero with

$$ \frac{d}{dx} Y_1(\hat{x}) = - \frac{\partial_x P(x, y)}{\partial_y P(x, y)} \Big|_{(x,y)=(\hat{x},Y_1(\hat{x}))} > 0; \quad (7.5) $$

iv) the function $x \mapsto Y_1(x) - Y_1(x_P^{**})$ does not vanish in the set $C(x_P^{**} - \delta, x_P^{**} + \delta)\setminus[x_P^{**}, x_P^{**} + \delta[$ for some $\delta > 0$, and as $x \to x_P^{**}$,

$$ Y_1(x) - Y_1(x_P^{**}) \sim -c\sqrt{x_P^{**} - x} \quad \text{and} \quad \frac{d}{dx} Y_1(x) \sim \frac{c}{2\sqrt{x_P^{**} - x}} \quad (7.6) $$

with

$$ c = \sqrt{\frac{\partial_x P(x, y)}{\partial_y^2 P(x, y)}} \Big|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))} > 0. $$

### 7.2.1 The main ideas and the sketch of the proof of Proposition 7.1

To prove this result, we first get a probabilistic representation of the function $Y_1$. As a consequence, we obtain that the function $Y_1$ is strictly convex on the line segment $[x_P^*, x_P^{**}]$ and can be continued as an analytic function to the open annulus $C(x_P^*, x_P^{**})$ and as a continuous function to the closed annulus $\bar{C}(x_P^*, x_P^{**})$ satisfying there the inequality (7.4) and the identity (7.3). This is a subject of the results of Subsection 7.2.2.

Next, in Subsection 7.2.3, we extend the functions $Y_1$ and $Y_2$ to a neighborhood of $x_P^{**}$ as two branches of a two-valued analytic function having a branching point $x_P^{**}$ and we get (7.6). This result is obtained by using the implicit function theorem and the Morse lemma.

Finally, in Subsection 7.2.4, the proof of Proposition 7.1 is completed: by using the implicit function theorem and relations (7.4) and (7.3) on the circle $\{x \in \mathbb{C} : |x| = x_P^{**}\}$, we extend the function $Y_1$ and the identity (7.3) to the whole set $U_\varepsilon = C(x_P^*, x_P^{**} + \varepsilon)\setminus[x_P^{**}, x_P^{**} + \varepsilon[$ for some $\varepsilon > 0$, and we prove the two last assertions of Proposition 7.1.

### 7.2.2 Probabilistic representation of the functions $Y_1$ its consequences

Consider the homogeneous random walk $(S(n) = (S_1(n), S_2(n))$ on $\mathbb{Z}^2$ with transition probabilities

$$ \mathbb{P}_j(S(1) = k) = \mu(k - j), \quad \forall k, j \in \mathbb{Z}^2, $$

and the first time $\tau_1$ when the random walk $(S(n))$ hits the set $\mathbb{Z} \times \{0\}$ :

$$ \tau_1 = \inf\{n \ge 1 : S(n) \in \mathbb{Z} \times \{0\}\}, $$

By Lemma 2.2 of [16]
**Lemma 7.1.** Under the hypotheses of Proposition 7.1, for any $j = (j_1, j_2) \in \mathbb{Z} \times \mathbb{N}^*$ and $x \in [x_P^*, x_P^{**}]$,

$$ \mathbb{E}_j(x^{S_1(\tau_1)}; \tau_1 < +\infty) = x^{j_1} Y_1(x)^{j_2}. $$

As a straightforward consequence of this statement one gets the following probabilistic representation of the function $Y_1$:

**Corollary 7.2.** Under the hypotheses (A1) and (A2), for any $x \in [x_P^*, x_P^{**}]$,

$$ Y_1(x) = \mathbb{E}_{(0,1)}(x^{S_1(\tau_1)}; \tau_1 < +\infty) \qquad (7.7) $$

With this result we get

**Lemma 7.2.** Under the hypotheses of Proposition 7.1, the function $Y_1$ is strictly convex on $[x_P^*, x_P^{**}]$ and can be extended to the set $\bar{C}(x_P^*, x_P^{**})$ as a function

$$ Y_1(x) = \sum_{k_1 \in \mathbb{Z}} \mathbb{P}_{(0,1)}(S(\tau_1) = (k_1, 0), \tau_1 < +\infty) x^{k_1} \qquad (7.8) $$

which is continuous on $\bar{C}(x_P^*, x_P^{**})$, analytic in $C(x_P^*, x_P^{**})$ and, on the set $\bar{C}(x_P^*, x_P^{**})$, satisfies the inequality (7.4) and the identity (7.3).

*Proof.* Indeed, by Corollary 7.2, the series

$$ \sum_{k_1 \in \mathbb{Z}} \mathbb{P}_{(0,1)}(S(\tau_1) = (k_1, 0), \tau_1 < +\infty) x^{k_1} = \mathbb{E}_{(0,1)}(x^{S_1(\tau_1)}; \tau_1 < +\infty) \qquad (7.9) $$

converges and the identity (7.8) holds for any real $x \in [x_P^*, x_P^{**}]$. Since the coefficients of the series (7.9) are real and non-negative, it follows that this series converges also on $\bar{C}(x_P^*, x_P^{**})$. Hence, by letting (7.8) for $x \in \bar{C}(x_P^*, x_P^{**})$, the function $Y_1$ can be extended to the set $\bar{C}(x_P^*, x_P^{**})$ as a function which is continuous on $\bar{C}(x_P^*, x_P^{**})$ and analytic in $C(x_P^*, x_P^{**})$.

To get (7.4) and to show that the function $Y_1$ is strictly convex on $[x_P^*, x_P^{**}]$, we show that all coefficients of the series (7.9) are strictly positive. For this we recall that under our hypotheses, the random walk ($S(n)$) is irreducible on $\mathbb{Z}^2$ and hence, for any $k_1 \in \mathbb{Z}$, there are $n \in \mathbb{N}$ and a sequence of points $\ell_0, \dots, \ell_n \in \mathbb{Z}^2$ with $\ell_0 = (0, 1)$ and $\sum_{i=0}^n \ell_i = (k_1, 0)$ such that

$$ \mu(\ell_i) > 0 \quad \text{for any } i \in \{1, \dots, n\}. $$

Moreover, without any restriction of generality, one can assume that for some $j \in \{1, \dots, n\}$, the second coordinate of each of the points $\ell_1, \dots, \ell_j$ is either zero or positive, and the second coordinate of each of the points $\ell_{j+1}, \dots, \ell_n$ strictly negative. Then for any $j \in \{0, \dots, n-1\}$, the second coordinate of the point $\ell_0 + \dots + \ell_n$ is strictly positive and consequently,

$$ \mathbb{P}_{(0,1)}(S(\tau_1) = (k_1, 0), \tau_1 < +\infty) \geq \prod_{i=1}^{n} \mu(\ell_i) > 0. $$

All coefficients of the series (7.9) are therefore strictly positive.

Since for any $k_1 \in \mathbb{Z}$, the function $x \mapsto \mathbb{P}_{(2,0)}(S(\tau_1) = (k_1, 0), \tau_1 < +\infty)x^{k_1}$ is convex on $[x_P^*, x_P^{**}]$ and the function $x \mapsto \mathbb{P}_{(2,0)}(S(\tau_1) = (2, 0), \tau_1 < +\infty)x^2$ is strictly convex on $[x_P^*, x_P^{**}]$, this proves that the function $Y_1$ is strictly convex on $[x_P^*, x_P^{**}]$. And using moreover Proposition P7.5 of [29] we get (7.4).

Remark finally that by relation (7.4), the function $x \mapsto (x, Y_1(x))$ maps the closed annulus $\bar{C}(x_P^*, x_P^{**})$ to the set

$$ \bar{\Gamma}_1 = \{(x, y) \in \mathbb{C}^2 : x \in \bar{C}(x_P^*, x_P^{**}), |y| \leq Y_1(|x|)\}. $$
Since under the hypotheses of Proposition 7.1, the function

$$Q(x, y) = xy - \sum_{k=(k_1, k_2) \in \mathbb{N}^2} \mu(k_1 - 1, k_2 - 1) x^{k_1} y^{k_2}$$

is analytic in a neighborhood of the set $\bar{\Gamma}_1$, and the function $Y_1$ is already extended as an analytic function to the open annulus $C(x_P^*, x_P^{**})$ and as a continuous function to the closed annulus $\bar{C}(x_P^*, x_P^{**})$, it follows that the function $x \mapsto Q(x, Y_1(x))$ is analytic in $C(x_P^*, x_P^{**})$ and continuous on $\bar{C}(x_P^*, x_P^{**})$. Since moreover the identity (7.3) holds for any real $x \in [x_P^*, x_P^{**}]$, by the uniqueness of the analytic continuation to the set $C(x_P^*, x_P^{**})$ and by continuity of the function $x \mapsto Q(x, Y_1(x))$ on $\bar{C}(x_P^*, x_P^{**})$, this proves that the identity (7.3) holds also on the whole set $\bar{C}(x_P^*, x_P^{**})$. □

### 7.2.3 Analytic continuation of the functions $Y_1$ and $Y_2$ to a neighborhood of the branching point $x_P^{**}$

Now, we extend the functions $Y_1$ and $Y_2$ to a neighborhood of $x_P^{**}$ as two branches of a two-valued analytic function having a branching point $x_P^{**}$ and we get (7.6).

**Lemma 7.3.** Under the hypotheses of Proposition 7.1, for some $\varepsilon > 0$ small enough, the functions $Y_1$ and $Y_2$ can be continued to the disk $B(x_P^{**}, \varepsilon)$ as two branches of two-valued analytic in $B(x_P^{**}, \varepsilon) \setminus \{x_P^{**}\}$ function such that for any $x \in B(x_P^{**}, \varepsilon) \setminus [x_P^{**}, x_P^{**} + \varepsilon]$,

$$P(x, Y_1(x)) = P(x, Y_2(x)) = 1, \qquad (7.10)$$

and there is an analytic in $B(0, \sqrt{\varepsilon})$ function $F_Y$ such that

$$Y_1(x) = F_Y(-\sqrt{x_P^{**} - x}), \qquad Y_2(x) = F_Y(\sqrt{x_P^{**} - x}) \qquad (7.11)$$

and

$$\left. \frac{d}{du} F_Y(u) \right|_{u=0} = \sqrt{\left. \frac{\partial_x P(x, y)}{\partial_{yy} P(x, y)} \right|_{(x,y)=(x_P^{**}, Y_1(x_P^{**})}} > 0. \qquad (7.12)$$

*Proof.* By the definition of the point $x_P^{**}$ and the functions $Y_1, Y_2$, we have

$$Y_1(x_P^{**}) = Y_2(x_P^{**}), \qquad P(x_P^{**}, Y_1(x_P^{**})) = 1,$$

and by Lemma 4.1,

$$\partial_y P(x, y)|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))} = 0, \quad \text{and} \quad \partial_x P(x, y)|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))} > 0. \qquad (7.13)$$

Hence, by the implicit function theorem, there are a neighborhood $\mathcal{V}$ of the point $x_P^{**}$ and an analytic in a neighborhood $U$ of $Y_1(x_P^{**})$ function $y \mapsto \psi(y)$ such that for any $(x, y) \in U \times V$,

$$P(x, y) = 1 \Leftrightarrow x = \psi(y),$$

$$\psi(Y_1(x_P^{**})) = x_P^{**}, \quad \left. \frac{d}{dy} \psi(y) \right|_{y=Y_1(x_P^{**})} = 0 \quad \text{and} \quad \left. \frac{d^2}{dy^2} \psi(y) \right|_{y=Y_1(x_P^{**})} = - \left. \frac{\partial^2 P(x, y)}{\partial_x P(x, y)} \right|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))} < 0,$$

where the last relation follows from the second relation of (7.13) because under the hypotheses (A1), the real valued function $y \mapsto P(x_P^{**}, y)$ is strictly convex in a neighborhood of $Y_1(x_P^{**})$.
Using the Morse lemma we conclude therefore that for some neighborhood $\tilde{U} \subset U$ of $Y_1(x_P^{**})$, there is a $\mathbb{C}$-diffeomorphism $\omega$ from $\tilde{U}$ to a neighborhood $\tilde{V}$ of 0, with

$$ \omega(Y_1(x_P^{**})) = 0 \quad \text{and} \quad \left. \frac{d}{dy} \omega(y) \right|_{y=Y_1(x_P^{**})} = \sqrt{\frac{\partial^2 P(x, y)}{\partial y^2 \partial_x P(x, y)}} \Big|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))} \qquad (7.14) $$

such that $\tilde{V} + x_P^{**} \subset V$ and for any $(x, y) \in (\tilde{V} + x_P^{**}) \times \tilde{U}$,

$$ P(x, y) = 1 \Leftrightarrow x = x_P^{**} - \omega^2(y). $$

Without any restriction of generality, one can assume that $\omega(\tilde{U}) = \tilde{V} = B(0, \sqrt{\epsilon})$ with $\epsilon > 0$ small enough. Then for $x \in B(x_P^{**}, \epsilon) \setminus [x_P^{**}, x_P^{**} + \epsilon]$, and $y \in \tilde{U}$, from the above relation it follows that

$$ P(x, y) = 1 \Leftrightarrow \left( \omega(y) = \sqrt{x_P^{**} - x} \quad \text{or} \quad \omega(y) = -\sqrt{x_P^{**} - x} \right) \\ \Leftrightarrow \left( y = \omega^{-1}(\sqrt{x_P^{**} - x}) \quad \text{or} \quad y = \omega^{-1}(-\sqrt{x_P^{**} - x}) \right) $$

with the square root function $\sqrt{\cdot}$ analytic in $\mathbb{C}\setminus[-\infty, 0]$. Since the inverse to $\omega$ function $\omega^{-1}$ is analytic in $B(0, \sqrt{\epsilon})$, and since for real $x \in [x_P^{**}, x_P^{**} + \epsilon]$ and $y > 0$,

$$ P(x, y) = 1 \Leftrightarrow (y = Y_1(x) \quad \text{or} \quad y = Y_2(x)) $$

we conclude therefore that the functions $Y_1$ and $Y_2$ can be extended to the set $B(x_P^{**}, \epsilon) \setminus [x_P^{**}, x_P^{**} + \epsilon]$ as analytic functions such that either

$$ Y_1(x) = \omega^{-1}(\sqrt{x_P^{**} - x}) \quad \text{and} \quad Y_2(x) = \omega^{-1}(-\sqrt{x_P^{**} - x}), \quad \forall x \in B(x_P^{**}, \epsilon) \setminus [x_P^{**}, x_P^{**} + \epsilon], $$

or

$$ Y_1(x) = \omega^{-1}(-\sqrt{x_P^{**} - x}) \quad \text{and} \quad Y_2(x) = \omega^{-1}(\sqrt{x_P^{**} - x}), \quad \forall x \in B(x_P^{**}, \epsilon) \setminus [x_P^{**}, x_P^{**} + \epsilon]. $$

These relations show that the function $\omega^{-1}$ is real valued on the interval $]-\sqrt{\epsilon}, \sqrt{\epsilon}]$, and hence, by the second relation of (7.14), it is strictly increasing in a neighborhood of 0. Since the function $Y_1$ is increasing on the interval $[X_1(y^*), x_P^{**}]$ and the function $Y_2$ is decreasing on the interval $[X_1(y^*), x_P^{**}]$, it follows that (7.10) and (7.11) hold with $F_Y = \omega^{-1}$. Relation (7.12) follows from (7.14). $\square$

### 7.2.4 Analytic continuation of the function $Y_1$ to the set $C(x_P^*, x_P^{**} + \epsilon) \setminus [x_P^{**}, x_P^{**} + \epsilon]$

Now we extend the function $Y_1$ and the identity (7.3) to the set $C(x_P^*, x_P^{**} + \epsilon) \setminus [x_P^{**}, x_P^{**} + \epsilon]$ for some small $\epsilon > 0$.

**Lemma 7.4.** Under the hypotheses of Proposition 7.1, for some $\epsilon > 0$ small enough, the function $x \mapsto Y_1(x)$ can be analytically continued to the set $C(x_P^*, x_P^{**} + \epsilon) \setminus [x_P^{**}, x_P^{**} + \epsilon]$, and satisfies there the identity (7.3).

*Proof.* By Lemma 7.2, the function $Y_1$ is already extended to the closed annulus $\bar{C}(x_P^*, x_P^{**})$ as an analytic function in $C(x_P^*, x_P^{**})$ and a continuous function on $\bar{C}(x_P^*, x_P^{**})$ satisfying there the identity (7.3), and by Lemma 7.3, for some $\epsilon > 0$, the function $Y_1$ is also already analytically continued to the set $B(x_P^{**}, \epsilon) \setminus [x_P^{**}, x^{**} + \epsilon]$. Hence, to prove this lemma, it is sufficient to show that the function $Y_1$ can be analytically continued to a neighborhood of the set $\{x \in \mathbb{C} : |x| = x_P^{**}, |x| \neq x_P^{**}\}$. In order to get this result, we use the implicit function theorem.
By Lemma 7.2, $Q(\tilde{x}, Y_1(\tilde{x})) = 0$ for any point $\tilde{x} \in \mathbb{C}$ with $|\tilde{x}| = x_P^{**}$ and under the hypotheses of Proposition 7.1, the function $Q$ is analytic in a neighborhood of $(\tilde{x}, Y_1(\tilde{x}))$. Hence, by the implicit function theorem, it is sufficient to show that for any such a point $\tilde{x}$,

$$ \partial_y Q(\tilde{x}, Y_1(\tilde{x})) \neq 0 \quad \text{whenever} \quad \tilde{x} \neq x_P^{**}. \qquad (7.15) $$

To get this relation, we let us remark that according to the definition (2.24) of the function $Q$ and using (7.4), for any point $\tilde{x} \in \mathbb{C}$ with $|\tilde{x}| = x_P^{**}$ and such that $\tilde{x} \neq x_P^{**}$, one has

$$ \begin{aligned} |\tilde{x} - \partial_y Q(\tilde{x}, Y_1(\tilde{x}))| &\le \sum_{k_1 \in \mathbb{Z}} \sum_{k_2=0}^{\infty} (k_2+1)\mu(k_1, k_2)(x_P^{**})^{k_1+1}|Y_1(\tilde{x})|^{k_2} \\ &< \sum_{k_1 \in \mathbb{Z}} \sum_{k_2=0}^{\infty} (k_2+1)\mu(k_1, k_2)(x_P^{**})^{k_1+1}Y_1(x_P^{**})^{k_2} = x_P^{**} - \partial_y Q(x_P^{**}, Y_1(x_P^{**})) \end{aligned} $$

where

$$ \partial_y Q(x_P^{**}, Y_1(x_P^{**})) = x_P^{**} - x_P^{**} P(x_P^{**}, Y_1(x_P^{**})) - x_P^{**} Y_1(x_P^{**}) \partial_y P x_P^{**}, Y_1(x_P^{**}) = 0 $$

because $P(x_P^{**}, Y_1(x_P^{**})) = 1$ and by Lemma 4.1, $\partial_y P(x_P^{**}, Y_1(x_P^{**})) = 0$. Hence, for any $\tilde{x} \in \mathbb{C}$ with $|\tilde{x}| = x_P^{**}$ and such that $\tilde{x} \neq x_P^{**}$,

$$ |\partial_y Q(\tilde{x}, Y_1(\tilde{x}))| \ge |\tilde{x}| - |\tilde{x} - \partial_y Q(\tilde{x}, Y_1(\tilde{x}))| = x_P^{**} - |\tilde{x} - \partial_y Q(\tilde{x}, Y_1(\tilde{x}))| > 0 $$

and consequently, (7.15) holds. $\square$

<figure><img src="image_4.png" /></figure>

### 7.2.5 Proof of Proposition 7.1

The first two assertions of Proposition 7.1 are proved by Lemma 7.2, Lemma 7.3 and Lemma 7.4.

To prove the third assertion of Proposition 7.1, recall that by Lemma 7.2, the function $Y_1$ is analytic in the annulus $C(x_P^*, x_P^{**})$ and satisfies there the inequality (7.4) and remark that any point $\hat{x} \in ]X_1(y_P^*), x_P^{**}[$ belongs to the annulus $C(x_P^*, x_P^{**})$ because $x_P^* < X_1(y_P^*) < x_P^{**}$. Hence, for any $\hat{x} \in ]X_1(y_P^*), x_P^{**}[$, the function $x \mapsto (Y_1(x) - Y_1(\hat{x}))$ is analytic in a neighborhood of the circle $\{x \in \mathbb{C} : |x| = \hat{x}$, and by (7.4), the point $\hat{x}$ is an only zero of this function in the circle $\{x \in \mathbb{C} : |x| = \hat{x}$. Since the zeros of analytic functions are always isolated, this proves that for any $\hat{x} \in ]X_1(y_P^*), x_P^{**}[$, there is $\delta > 0$ for which the function $x \mapsto (Y_1(x) - Y_1(\hat{x}))$ does not vanish in the set $C(\hat{x} - \delta, \hat{x} + \delta) \setminus \{\hat{x}\}$.

Moreover, we have

- by Lemma 4.1, the real valued function $Y_1$ is strictly increasing on the interval $]X_1(y_P^*), x_P^{**}[$,

- by Lemma 7.2, it is also strictly convex on $]X_1(y_P^*), x_P^{**}[$,

- and by Lemma 4.1, $P(\hat{x}, Y_1(\hat{x})) = 1$ and $\partial_y P(\hat{x}, Y_1(\hat{x})) < 0$.

Hence, for any $\hat{x} \in ]X_1(y_P^*), x_P^{**}[$, using the implicit function theorem, one gets (7.5). From this relation, it follows moreover that for any $\hat{x} \in ]X_1(y_P^*), x_P^{**}[$, the point the point $\hat{x}$ is a simple zero of the function $x \mapsto Y_1(x) - Y_1(\hat{x})$. The third assertion of Proposition 7.1 is therefore also proved.

To get the last assertion of Proposition 7.1, recall that by Lemma 7.3 and Lemma 7.4, the function $Y_1$ was already extended to the set $U_\epsilon = C(x_P^*, x^{**} + \epsilon)\setminus]x_P^{**}, x_P^{**} + \epsilon[$ as an analytic function in $\tilde{U}_\epsilon = C(x_P^*, x^{**} + \epsilon)\setminus[x_P^{**}, x_P^{**} + \epsilon[$ and a continuous function on $U_\epsilon$ satisfying there
the inequality (7.4). The function $x \mapsto Y_1(x) - Y_1(x_P^{**})$ is therefore analytic in $\tilde{U}_\varepsilon$ and continuous on $U_\varepsilon$, and moreover by (7.4), the point $x_P^{**}$ is its only zero in the circle $\{x \in \mathbb{C} : |x| = x_P^{**}\}$. To complete the proof of the fourth assertion of Proposition 7.1, it is therefore sufficient to get (7.6) and to show that the point $x_P^{**}$ is an isolated zero of the function $x \mapsto Y_1(x) - Y_1(x_P^{**})$. For this, we remark that by Lemma 7.3, as $x \to x_P^{**}$,

$$Y_1(x) - Y_1(x_P^{**}) = F_Y(-\sqrt{x_P^{**} - x}) - F_Y(0) \sim -c\sqrt{x_P^{**} - x} \quad (7.16)$$

and

$$\frac{d}{dx}Y_1(x) \sim \frac{c}{2\sqrt{x_P^{**} - x}}$$

with

$$c = \left. \frac{d}{du} F_Y(0) \right|_{u=0} = \sqrt{\partial_x P(x, y)/\partial_{yy} P(x, y)} \Big|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))} > 0.$$

Relations (7.6) are therefore proved, and since the constant $c$ is strictly positive, from (7.16) it follows that the point $x_P^{**}$ is an isolated zero of the function $x \mapsto Y_1(x) - Y_1(x_P^{**})$.

### 7.3 The properties of the mapping $x \mapsto (x, Y_1(x))$ and their consequences

Remark that by (7.4), the function $x \mapsto (x, Y_1(x))$ maps the closed annulus to the set $\{(x, y) \in \mathbb{C}^2 : x \in C(x_P^{**}, x_P^{**}), |y| \le Y_1(|x|)\}$, and under the hypotheses (A1)-(A3), for any $j \in \mathbb{Z}_+^2$, the functions $(x, y) \mapsto \phi_1(x, y)$, $(x, y) \mapsto \psi_2(x, y)$ and $(x, y) \mapsto L_j(x, y)$ are analytic in a neighborhood of this set. Hence, as a straightforward consequence of Proposition 7.1 one gets **Corollary 7.3.** Under the hypotheses (A1)-(A3), for some $\delta > 0$, the functions $x \mapsto \phi_1(x, Y_1(x))$, $x \mapsto \psi_2(x, Y_1(x))$ and $x \mapsto L_j(x, Y_1(x))$ for any $j \in \mathbb{Z}_+^2$, are analytic in the set $\tilde{U}_\delta$ and continuous on the set $U_\delta$.

Another consequence of Proposition 7.1 is the following property of the mapping $x \mapsto (x, Y_1(x))$.

**Proposition 7.4.** Under the hypotheses (A1)-(A4), for some $\delta > 0$, the function $x \mapsto (x, Y_1(x))$ maps the annulus $C(x_d - \delta, x_d)$ to the set $\{(x, y) \in \Omega(\Gamma) : |x| < x_d, |y| < y_d\}$. Moreover, for any neighborhood $V$ of the set $\Omega(\Gamma)$ in $\mathbb{C}^2$ and $\hat{y} > Y_1(x_d)$, there is $\delta > 0$, for which the function $x \mapsto (x, Y_1(x))$ maps the set $C(x_d - \delta, x_d + \delta) \cap U_\delta$ to the set $\{(x, y) \in V : |y| < \hat{y}\}$.

*Proof.* Remark that by (7.4), for any $x \in]x_P^{**}, x_P^{**}]$,

$$|Y_1(x)| \le Y_1(|x|) < Y_2(|x|).$$

Hence, the function $x \mapsto (x, Y_1(x))$ maps the set $C(x_P^{**}, x_P^{**})$ to the set

$$\Gamma_1 = \{(x, y) \in \mathbb{C}^2 : x \in C(x_P^{**}, x_P^{**}), |y| < Y_2(|x|)\}.$$

For any $x \in]x_P^{**}, x_P^{**}]$, $Y_1(x)$ and $Y_2(x)$ are the only real and positive solution of the equation $P(x, y) = 1$, $Y_1(x) < Y_2(x)$ and for any $Y_1(x) < y < Y_2(x)$, the point $(x, y)$ belongs to the interior of the set $D = \{(x, y) \in [0, +\infty]^2 : P(x, y) \le 1\}$ (see Lemma 4.1 for more details). Hence for any $(x, y) \in \Gamma_1$, there is a point $(x', y') \in D$ such that $|x| < x'$ and $|x| < y'$ and consequently, the set $\Gamma_1$ is included to the set $\Gamma$. The mapping $x \mapsto (x, Y_1(x))$ maps therefore the set $C(x_P^{**}, x_P^{**})$ to the set $\Gamma$.

Consider now the case when one of the cases (B0)-(B4) holds. In this case, (see Proposition 2.2 and relations (2.33), (2.34)), we have $Y_1(x_d) < y_d$. Since the function $x \mapsto (x, Y_1(x))$ maps the annulus $C(x_P^{**}, x_P^{**})$ to the set $\Gamma$ and by Proposition 7.1, the function
$Y_1$ is continuous on $U_\varepsilon$, it follows that for some $\delta > 0$, the function $x \mapsto (x, Y_1(x))$ maps the annulus $C(x_d - \delta, x)$ to the set $\{(x, y) \in \Omega(\Gamma) : |x| < x_d, |y| < y_d\}$.

When one of the cases (B0)-(B4) holds, the first assertion of Proposition 7.4 is therefore proved.

Consider now the case when (B5) or (B6) holds. In this case (see Proposition 2.2 and relations (2.33), (2.34)), the point $(x_d, y_d) = (X_2(y^{**}), y^{**})$ belongs to the curve $S_{21} = \{(x, Y_1(x)) : x \in [X_1(y_P^*), x_P^{**}]\} = \{(X_2(y), y) : y \in [y_P^*, Y_1(x_P^{**})]\}$ and $y_P^* \le y^* < y_d = y^{**} < y_P^{**}$. Since by lemma 4.1, the functions $X_2 : [y_P^*, Y_1(x_P^{**})] \to [X_1(y_P^*), x_P^{**}]$ and $Y_1 : [X_1(y_P^*), x_P^{**}] \to [y_P^*, Y_1(x_P^{**})]$ are strictly increasing and inverse to each other, it follows that in this case, we have $x_P^* < X_2(y_d) = x_d$. Hence, for any $x \in C(x_P^*, x_d)$, using (7.4) one gets

$$|Y_1(x)| \le Y_1(|x|) < Y_1(x_d).$$

Since we have already proved that the function $x \mapsto (x, Y_1(x))$ maps the annulus $C(x_P^*, x_P^{**})$ to the set $\Gamma$, the last relations prove that for $\delta = x_d - x_P^* > 0$, this function maps the annulus $C(x_d - \delta, x_d)$ to the set $\{(x, y) \in \Omega(\Gamma) : |x| < x_d, |y| < y_d\}$.

In the case when one of the cases (B5) or (B6) holds, the first assertion of Proposition 7.4 is therefore also proved.

The second assertion of Proposition 7.4 holds because the function $x \mapsto (x, Y_1(x))$ is continuous in the set $U_\varepsilon$ including the closed annulus $\bar{C}(x_P^*, x_P^{**})$ and maps $\bar{C}(x_P^*, x_P^{**})$ to the closure of the set $\Gamma$.

□

Remark that under the hypotheses (A1)(ii), (A2) and (A3) (ii), (iv), the functions $(x, y) \mapsto Q(x, y) = xy(1-P(x, y))$, $(x, y) \mapsto \psi_1(x, y) = x(\phi_1(x, y) - 1)$ and $(x, y) \mapsto \psi_2(x, y) = y(1-\phi_2(x, y))$ can be continued as analytic functions to some neighborhood of the set $\Omega(\bar{\Gamma})$. Hence, from Theorem 2.1 it follows

**Corollary 7.5.** Under the hypotheses (A1)-(A4), there is a neighborhood $\mathcal{V} \subset \mathbb{C}^2$ of the set $\Omega(\bar{\Gamma})$ such that for any $j \in \mathbb{Z}^2$,

- the function $(x, y) \mapsto \psi_1(x, y)h_{1j}(x)$ is analytic in the set $\{(x, y) \in \mathcal{V} : |x| < x_d\}$;

- the function $(x, y) \mapsto \psi_2(x, y)h_{2j}(y)$ is analytic in the set $\{(x, y) \in \mathcal{V} : |y| < y_d\}$

- the function $(x, y) \mapsto R_j(x, y) = Q(x, y)h_j(x, y)$ can be extended as an analytic function to the set $\{(x, y) \in \mathcal{V} : |x| < x_d, |y| < y_d\}$ by letting

$$R_j(x, y) = L_j(x, y) + \psi_1(x, y)h_{1j}(x) + \psi_2(x, y)h_{2j}(y). \quad (7.17)$$

When combined together, Proposition 7.4, Corollary 7.3 and Corollary 7.5 imply the following statement.

**Corollary 7.6.** Under the hypotheses (A1)-(A4) for some $\delta > 0$, and any $j \in \mathbb{Z}_+^2$,

i) the functions $x \mapsto -\psi_1(x, Y_1(x))h_{1j}(x) = (1 - \phi_1(x, Y_1(x)))H_j(x, 0)$ and $x \mapsto \psi_2(x, Y_1(x))h_{2j}(Y_1(x))$ are analytic in the annulus $C(x_d - \delta, x_d)$ and satisfy there the identity

$$(1 - \phi_1(x, Y_1(x)))H_j(x, 0) = L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x)); \quad (7.18)$$

ii) moreover, if one of the cases (B0)-(B4) holds with $x_d < x_P^{**}$ (i.e. if either one of the cases (B0), (B1), (B3) or (B4) holds or (B2) and $x_d < x_P^{**}$ hold), then for some $\delta > 0$, the function $x \mapsto L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x))$ is analytic in the annulus $C(x_d - \delta, x_d + \delta)$ and by (7.18), the function $x \mapsto \eta_j(x) = (1 - \phi_1(x, Y_1(x)))H_j(x, 0)$ can be extended as an analytic function to $C(x_d - \delta, x_d + \delta)$.
iii) if (B2) holds with $x_d = x_P^{**}$, then for some $\delta > 0$, the function $x \mapsto L_j(x, Y_1(x)) + \psi(x, Y_1(x))h_{2j}(Y_1(x))$ is analytic in the set $C(x_d - \delta, x_d + \delta) \cap \tilde{U}_\epsilon$ and continuous on the set $C(x_d - \delta, x_d + \delta) \cap U_\epsilon$, and by (7.18), the function $x \mapsto \eta_j(x) = (1 - \phi_1(x, Y_1(x)))H_j(x, 0)$ can be extended to the set $C(x_d - \delta, x_d + \delta) \cap U_\epsilon$ as an analytic function in $C(x_d - \delta, x_d + \delta) \cap \tilde{U}_\epsilon$ and a continuous function on $C(x_d - \delta, x_d + \delta) \cap U_\epsilon$.

With this result, when one of the cases (B0)-(B4) holds, the function $x \mapsto (1 - \phi_1(x, Y_1(x)))H_j(x, 0)$ is therefore already extended beyond the point $x_d$. To extend the function $x \mapsto H_j(x, 0)$ beyond the point $x_d$ we need to investigate the function $x \mapsto \phi_1(x, Y_1(x))$. This is a subject of the next section.

## 7.4 Analytic continuation and properties of the function $x \mapsto \phi_1(x, Y_1(x))$

By Corollary 7.3, for some $\delta > 0$, the function $x \mapsto \phi_1(x, Y_1(x))$ is already extended to the set $U_\delta$ as an analytic function in the set $\tilde{U}_\delta$ and a continuous function on the set $U_\delta$. We will need moreover the following properties of the function $x \mapsto \phi_1(x, Y_1(x))$.

**Proposition 7.7.** Under the hypotheses (A1)-(A3), the following assertions hold:

i) the function $x \mapsto \phi_1(x, Y_1(x))$ is strictly convex on the line segment $[x_P^*, x_P^{**}]$.

ii) for any $x \in C(x^*, x^{**})$,

$$ |\phi_1(x, Y_1(x))| \le \phi(|x|, Y_1(|x|)) < 1; \qquad (7.19) $$

iii) if $x^{**} < x_P^{**}$, then for some $\delta > 0$, the point $x^{**}$ is an only and simple zero of the function $x \mapsto 1 - \phi_1(x, Y_1(x))$ in the annulus $C(x^*, x^{**} + \delta)$ and

$$ \left. \frac{d}{dx} \phi_1(x, Y_1(x)) \right|_{x=x^{**}} > 0; \qquad (7.20) $$

iv) if $x^{**} = x_P^{**}$, then $\phi_1(x_P^{**}, Y_1(x_P^{**})) \le 1$;

v) if $x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$, then for some $\delta > 0$, the function $x \mapsto 1 - \phi_1(x, Y_1(x))$ has no zeros in $U_\delta = C(x^*, x_P^{**} + \delta) \setminus [x_P^{**}, x_P^{**} + \delta]$;

vi) if $x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) = 1$, then for some $\delta > 0$, the point $x_P^{**}$ is an only zero of the function $x \mapsto 1 - \phi_1(x, Y_1(x))$ in $U_\delta$ and as $x \to x_P^{**}$,

$$ 1 - \phi_1(x, Y_1(x)) \sim c\sqrt{x_P^{**} - x} \quad \text{with} \quad c = \partial_y\phi_1(x, y)\sqrt{\frac{\partial_x P(x, y)}{\partial_{yy} P(x, y)}}\bigg|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))} > 0. \qquad (7.21) $$

### 7.4.1 Outline of the proof of Proposition 7.7

To prove this proposition, we first get a probabilistic representation of the function $x \mapsto \phi_1(x, Y_1(x))$ similar to those of the function $x \mapsto Y_1(x)$. This is a subject of Corollary 7.8. In Section 7.4.3, as a straightforward consequence of this result, we get the first and the second assertion of Proposition 7.7. The proof of the third assertion of this statement is given in Section 7.4.4 and the proofs of the last tree assertions are completed in Section 7.4.5.

### 7.4.2 Probabilistic representation of the function $x \mapsto \phi_1(x, Y_1(x))$

The following probabilistic representation of the function $x \mapsto \phi_1(x, Y_1(x))$ is another useful for our purpose consequence of Lemma 7.1:
**Corollary 7.8.** Let $(\hat{S}(n))$ be a random walk on the half-plane $\mathbb{Z} \times \mathbb{N}$ with transition probabilities

$$ \mathbb{P}_k(\hat{S}(1) = j) = \begin{cases} \mu(j - k) & \text{for all } j, k = (k_1, k_2) \in \mathbb{Z} \times \mathbb{N} \text{ with } k_2 > 0 \\ \mu_1(j - k) & \text{for all } j, k = (k_1, k_2) \in \mathbb{Z} \times \mathbb{N} \text{ with } k_2 = 0 \end{cases} \quad (7.22) $$

and let $T_1$ be the first time when the random walk $(\hat{S}(n) = (\hat{S}_1(n), \hat{S}_2(n)))$ hits the boundary $\mathbb{Z} \times \{0\}$:

$$ T_1 = \inf\{n > 0 : \hat{S}(n) \in \mathbb{Z} \times \{0\}\}. $$

Then under the hypotheses (A1) - (A3), for any $x \in \bar{C}(x_P^*, x_P^{**})$

$$ \phi_1(x, Y_1(x)) = \mathbb{E}_{(0,0)} (x^{\hat{S}_1(T_1)}; T_1 < +\infty). \quad (7.23) $$

*Proof.* To get (7.23) from Lemma 7.1 it is sufficient to notice that by the Markov property, and according to the definition of the random walks $(S(n))$ and $(\hat{S}(n))$,

$$ \begin{aligned} \mathbb{E}_{(0,0)} (x^{\hat{S}_1(\tau_1)}; \tau_1 < +\infty) &= \sum_{k \in \mathbb{Z} \times \{0\}} \mu_1(k) x^{k_1} + \sum_{k=(k_1, k_2) \in \mathbb{Z} \times \mathbb{N}: k_2 \neq 0} \mu_1(k) \mathbb{E}_k(x^{S_1(T_1)}), \quad T_1 < +\infty \\ &= \sum_{k \in \mathbb{Z} \times \{0\}} \mu_1(k) x^{k_1} + \sum_{k=(k_1, k_2) \in \mathbb{Z} \times \mathbb{N}: k_2 \neq 0} \mu_1(k) x^{j_1} Y_1(x)^{j_2} = \phi_1(x, Y_1(x)). \end{aligned} \quad \square $$

### 7.4.3 Proof of the first and the second assertions of Proposition 7.7

With Corollary 7.8, by using the same arguments as in the proof of Lemma 7.2 we get

**Corollary 7.9.** Under the hypotheses (A1) - (A3), the function $x \mapsto \phi_1(x, Y_1(x))$ is strictly convex on $[x_P^*, x_P^{**}]$ and satisfies on the set $\bar{C}(x_P^*, x_P^{**})$ the following relation

$$ |\phi_1(x, Y_1(x))| < \phi_1(|x|, Y_1(|x|)), \quad \forall x \neq |x|. \quad (7.24) $$

Remark that this statement proves the first assertion of Proposition 7.7. Moreover, since by Corollary 4.1,

$$ \phi_1(x, Y_1(x)) < 1 \quad \text{for any } x \in ]x^*, x^{**}[, \quad (7.25) $$

using (7.24) one gets (7.19), and consequently, the second assertion of Proposition 7.7 also holds.

### 7.4.4 Proof of the third assertion of Proposition 7.7

Suppose now that $x^{**} < x_P^{**}$. In this case, by Corollary 4.1,

$$ \phi_1(x^{**}, Y_1(x^{**})) = 1 \quad \text{and} \quad \phi_1(x, Y_1(x)) > 1 \quad \text{for all} \quad x \in ]x^{**}, x_P^{**}]. \quad (7.26) $$

Using this relation together with (7.19) one get that the point $x^{**}$ is an only zero of the function $x \mapsto \phi_1(x, Y_1(x)) - 1$ in the set $\{x \in \mathbb{C} : x^* < |x| \le x^{**}\}$, and since the function $x \mapsto \phi_1(x, Y_1(x)) - 1$ is analytic in the neighborhood $\hat{U}_\delta$ of this set, it follows that for some $\hat{\delta} > 0$, the point $x^{**}$ is an only zero of this function in the annulus $C(x^*, x^{**} + \hat{\delta})$. Moreover, since the function $x \mapsto \phi_1(x, Y_1(x))$ is strictly convex on the line segment $[x_P^*, x_P^{**}]$ and by (2.9), in the case when $x^{**} < x_P^{**}$, the point $x^{**}$ belongs to the interior of this line segment, from (7.25) and (7.26) it follows that

$$ \left. \frac{d}{dx} \phi_1(x, Y_1(x)) \right|_{x=x^{**}} > 0, $$

and consequently, the point $x^{**}$ is in this case a simple zero of the function $x \mapsto \phi_1(x, Y_1(x)) - 1$. The third assertion of Proposition 7.7 is therefore also proved.
### 7.4.5 Proof of the last three assertions of Proposition 7.7

Suppose now that $x^{**} = x_P^{**}$. Then by Corollary 4.1, $\phi_1(x_P^{**}, Y_1(x_P^{**})) \le 1$, and consequently, the fourth assertion of Proposition 7.7 holds. Moreover, if $x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$, using (7.24) we get

$$|\phi_1(x, Y_1(x))] \le \phi_1(|x|, Y_1(|x|)) < 1, \text{ for any } x \in \mathbb{C} \text{ with } x^* < |x| \le x_P^{**},$$

and consequently, the function $x \mapsto \phi_1(x, Y_1(x)) - 1$ has no zeros in $\{x \in \mathbb{C} : x^* < |x| \le x_P^{**}\}$. Since the function $x \mapsto \phi_1(x, Y_1(x))$ is continuous on the set $U_\delta$, we conclude therefore that, for some $\hat{\delta} > 0$, it has no zeros in $U_\delta$.

Finally, if $x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) = 1$, using again (7.24) one gets that the point $x_P^{**}$ is an only zero of the function $x \mapsto \phi_1(x, Y_1(x)) - 1$ in the set $\{x \in \mathbb{C} : x^* < |x| \le x_P^{**}\}$. Moreover, since under our hypotheses, the function $(x, y) \mapsto \phi_1(x, y)$ is analytic in a neighborhood of the point $(x_P^{**}, Y_1(x_P^{**}))$ and since because of Assumption (A3)(iv),

$$\partial_y \phi_1(x_P^{**}, Y_1(x_P^{**})) > 0,$$

using (7.6) one gets that as $x \to x_P^{**}$,

$$\phi_1(x, Y_1(x)) - 1 \sim (x - x_P^{**}) \partial_x \phi_1(x_P^{**}, Y_1(x_P^{**})) - c\sqrt{x_P^{**} - x} \partial_y \phi_1(x_P^{**}, Y_1(x_P^{**})) \\ \sim -c\sqrt{x_P^{**} - x} \partial_y \phi_1(x_P^{**}, Y_1(x_P^{**}))$$

with

$$c = \sqrt{\partial_x P(x, y)/\partial_{yy}^2 P(x, y)} \Big|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))} > 0.$$

This proves that the point $x_P^{**}$ is in this case an isolated zero of the function $x \mapsto \phi_1(x, Y_1(x)) - 1$ in $U_\delta$. Since this function is continuous in a neighborhood $U_\delta$ of the set $\{x \in \mathbb{C} : x^* < |x| \le x_P^{**}\}$, and has no zeros in $\{x \in \mathbb{C} : x^* < |x| \le x_P^{**}, x \ne x_P^{**}\}$, we conclude therefore that for some $\hat{\delta} > 0$, the point $x_P^{**}$ is an only zero of the function $x \mapsto \phi_1(x, Y_1(x)) - 1$ in $U_\delta$. The last assertion of Proposition 7.7 is therefore also proved.

## 7.5 Analytic continuation of the function $x \mapsto H_j(x, 0)$, cases (B0)-(B4)

In this section, we extend the function $x \mapsto H_j(x, 0)$ beyond the point $x_d$ in the cases (B0)-(B4). The cases (B0), (B1), (B3), (B4), and the case (B2) with $x^{**} < x_P^{**}$ are considered in Proposition 7.10. The case (B2) with $x_d = x_P^{**}$ is considered in Proposition 7.11 below.

**Proposition 7.10.** Suppose that the conditions (A1)-(A3) are satisfied and let either one of the cases (B0), (B1), (B3), (B4) holds or (B2) and $x^{**} < x_P^{**}$ hold. Then for some $\delta > 0$ and any $j \in \mathbb{Z}_+$, the function $x \mapsto H_j(x, 0)$ can be extended as an analytic function to the set $B(0, x_d + \delta) \setminus \{x_d\}$, and (2.40) holds with $\varkappa_1(j)$ defined by (2.35) and $a_1 > 0$ defined by (2.41).

*Proof.* Under the hypotheses of this proposition, by Corollary 7.6, for some $\delta_1 > 0$, the function $x \mapsto L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x))$ is analytic in the annulus $C(x_d - \delta_1, x_d + \delta_1)$, and using the identities

$$(1 - \phi_1(x, Y_1(x)))H_j(x, 0) = -\psi_1(x, Y_1(x))h_{1j}(x) \qquad (7.27)$$

$$= L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x)) \qquad (7.28)$$

the function $x \mapsto (1 - \phi_1(x, Y_1(x)))H_j(x, 0)$ can be analytically continued to the open annulus $C(x_d - \delta_1, x_d + \delta_1)$. Since by Proposition 7.7, for some $\delta_2 > 0$, the function $x \mapsto (1 - \phi_1(x, Y_1(x)))^{-1}$
is analytic in the set $C(x_d - \delta_2, x_d + \delta_2) \setminus \{x_d\}$ and has at the point $x_d$ a simple pole with

$$ \lim_{x \to x_d} \frac{x_d - x}{1 - \phi_1(x, Y_1(x))} = \left( \left. \frac{d}{dx} \phi_1(x, Y_1(x)) \right|_{x=x_d} \right)^{-1} > 0, $$

we conclude therefore that the function $x \mapsto H_j(x, 0)$ can be extended as an analytic function to the set $B(0, x_d + \delta) \setminus \{x_d\}$, and that (2.40) holds with $\varkappa_1(j)$ given by (2.35), and $a_1 > 0$ given by (2.41). $\square$

**Proposition 7.11.** Suppose that the conditions (A1)-(A3) are satisfied and let (B2) and $x_d = x_P^{**}$ hold. Then the following assertions holds.

- If $\phi_1(x_P^{**}, Y_1(x_P^{**})) = 1$, then for some $\delta > 0$, the function $x \mapsto H_j(x, 0)$ can be extended as an analytic function to the set $B(0, x_P^{**} + \delta) \setminus [x_P^{**}, x_P^{**} + \delta]$ satisfying (2.42) with $\varkappa_1(j)$ defined by (2.35) with $x_d = x_P^{**}$ and $a_2 > 0$ defined by (2.43).

- If $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$, then for some $\delta > 0$, the function $x \mapsto H_j(x, 0)$ can be extended to the set $B(0, x_P^{**} + \delta) \setminus [x_P^{**}, x_P^{**} + \delta]$ as an analytic function in $B(0, x_P^{**} + \delta) \setminus [x_P^{**}, x_P^{**} + \delta]$ and a continuous function on $B(0, x_P^{**} + \delta) \setminus [x_P^{**}, x_P^{**} + \delta]$, satisfying (2.44) with $\tilde{\varkappa}_1(j)$ defined by (2.36) and $\tilde{a}_2 > 0$ defined by (2.45).

*Proof.* Indeed, in this case, by Corollary 7.6, for some $0 < \delta_1 \le \varepsilon$, the function $x \mapsto L_j(x, Y_1(x)) + \psi(x, Y_1(x))h_{2j}(Y_1(x))$ is analytic in $C(x_d - \delta_1, x_d + \delta_1) \cap \tilde{U}_\varepsilon$ and continuous on $C(x_d - \delta_1, x_d + \delta_1) \cap U_\varepsilon$, and using the identities (7.27) end (7.28), the function $x \mapsto \eta_j(x) = (1 - \phi_1(x, Y_1(x)))H_j(x, 0)$ can be extended to the set $C(x_d - \delta_1, x_d + \delta_1) \cap U_\varepsilon$ as an analytic function in $C(x_d - \delta_1, x_d + \delta_1) \cap \tilde{U}_\varepsilon$ and a continuous function on $C(x_d - \delta_1, x_d + \delta_1) \cap U_\varepsilon$. Since by Proposition 7.7, for some $0 < \delta \le \delta_1$, the function $x \mapsto 1/(1 - \phi_1(x, Y_1(x)))$ is analytic in the set $C(x_d - \delta, x_d + \delta) \cap \tilde{U}_\varepsilon$, we get therefore that the function $x \mapsto H_j(x, 0)$ can be extended as an analytic function to the set $C(x_d - \delta, x_d + \delta) \cap \tilde{U}_\varepsilon$ by letting

$$ \begin{aligned} H_j(x, 0) &= \eta_j(x)(1 - \phi_1(x, Y_1(x)))^{-1} \\ &= (L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x))) (1 - \phi_1(x, Y_1(x)))^{-1}. \end{aligned} \quad (7.29) $$

Recall moreover that by Proposition 7.7, when $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$, the function $x \mapsto 1/(1 - \phi_1(x, Y_1(x)))$ is continuous on $C(x_d - \delta, x_d + \delta) \cap U_\varepsilon$, and when $\phi_1(x_P^{**}, Y_1(x_P^{**})) = 1$,

$$ \lim_{x \to x_P^{**}} \frac{\sqrt{x_P^{**} - x}}{1 - \phi_1(x, Y_1(x))} = \left. \left( \partial_y \phi_1(x, y) \sqrt{\partial_x P(x, y)/\partial_{yy}^2 P(x, y)} \right)^{-1} \right|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))} > 0. \quad (7.30) $$

Hence, in the case when $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$, the function $x \mapsto H_j(x, 0)$ can be extended as a continuous function to the set $C(x_d - \delta, x_d + \delta) \cap U_\varepsilon$, and in the case when $\phi_1(x_P^{**}, Y_1(x_P^{**})) = 1$, using (7.29) (7.30), one gets (2.42) with $\varkappa_1(j)$ defined by (2.35), and $a_2 > 0$ given by (2.43).

Remark finally that when $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$, using (7.29), for $x \in B(0, x_P^{**})$ closed enough to $x_P^{**}$, one gets

$$ \begin{aligned} \frac{d}{dx} H_j(x, 0) &= \frac{d}{dx} (r_j(x, Y_1(x))(1 - \phi_1(x, Y_1(x)))^{-1}) \\ &= \frac{d}{dx} ((L_j(x, Y_1(x)) + (\phi_2(x, Y_1(x)) - 1)H_j(0, Y_1(x))(1 - \phi_1(x, Y_1(x)))^{-1}) \\ &= \partial_x ((L_j(x, y) + (\phi_2(x, y) - 1)H_j(0, y))(1 - \phi_1(x, y))^{-1}) \Big|_{y=Y_1(x)} \\ &\quad + \partial_y ((L_j(x, Y_1(x)) + (\phi_2(x, Y_1(x)) - 1)H_j(0, Y_1(x))(1 - \phi_1(x, Y_1(x)))^{-1}) \Big|_{y=Y_1(x)} \times \frac{d}{dx} Y_1(x) \end{aligned} $$
where by (7.6), as $x \to x_P^{**}$,

$$ \frac{d}{dx} Y_1(x) \sim \frac{c}{2\sqrt{x_P^{**} - x}} \quad \text{with} \quad c = \frac{1}{2} \sqrt{\partial_x P(x, y) / \partial_{yy} P(x, y)} \Big|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))} > 0. $$

Since in the case when $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$, the function

$$ (x, y) \mapsto (L_j(x, y) + (\phi_2(x, y) - 1)H_j(0, y))(1 - \phi_1(x, y))^{-1} $$

is analytic in a neighborhood of the point $(x_P^{**}, Y_1(x_P^{**}))$, it follows (2.44) with $\tilde{a}_2 > 0$ given by (2.45). $\square$

## 7.6 Analytic continuation of the function $x \mapsto H_j(x, 0)$, cases (B5) and (B6)

Now we are ready to extend the function $x \mapsto H_j(x, 0)$ beyond the point $x_d$ when either (B5) or (B6) holds. This is a subject of Proposition 7.12 below.

**Proposition 7.12.** Under the hypotheses (A1)-(A3), there is $\delta > 0$ such that for any $j \in \mathbb{Z}_+^2$, the following assertions hold

i) If (B6) holds, then the function $x \mapsto H_j(x, 0)$ can be extended as an analytic function to the set $B(0, x_d + \delta) \setminus \{x_d\}$ and (2.52) holds with $a_5 > 0$ given by (2.53).

ii) If (B5) holds and $x_d = x^{**} < x_P^{**}$, then the function $x \mapsto H_j(x, 0)$ can be extended as an analytic function to the set $B(0, x_d + \delta) \setminus \{x_d\}$ and (2.46) holds with $a_3 > 0$ given by (2.47).

iii) If (B5), $x_d = x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) = 1$ hold, the function $x \mapsto H_j(x, 0)$ can be extended as an analytic function to the set $B(0, x_d + \delta) \setminus [x_d, x_d + \delta]$ and (2.48) holds with $a_4 > 0$ given by (2.49).

iv) If (B5), $x_d = x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$ hold, the function $x \mapsto H_j(x, 0)$ can be extended as an analytic function to the set $B(0, x_d + \tilde{\delta}) \setminus [x_d, x_d + \tilde{\delta}]$ and (2.50) holds with $\tilde{a}_4$ given by (2.51).

Recall that under our hypotheses, all functions $(x, y) \mapsto L_j(x, y)$, $(x, y) \mapsto \psi_1(x, y)$ and $(x, y) \mapsto \psi_2(x, y)$ are analytic in some neighborhood $\mathcal{V}$ of the set $\Omega(\bar{\Gamma})$. Throughout this section, the set $\mathcal{V}$ will be given.

### 7.6.1 Preliminary result

We begin the proof of this proposition with the following lemma.

**Lemma 7.5.** Under the hypotheses (A1)-(A3) and if one of the cases (B5) or (B6) holds, there is $\delta > 0$ such that for any $j \in \mathbb{Z}_+^2$, the function $x \mapsto (1 - \phi_1(x, Y_1(x)))H_j(0, Y_1(x))$ can be analytically continued to the set $C(x_d - \delta, x_d + \delta) \cap U_\varepsilon \setminus \{x_d\}$.

*Proof.* Suppose that either (B5) or (B6) holds. Then by Proposition 2.2 and using relations (2.33), (2.34)), one gets

$$ X_1(y_d) < x_d = X_2(y_d) \le x^{**}, \quad y^* < y_d = y^{**} = Y_1(x_d) < y_P^{**} \quad \text{and} \quad (x_d, y_d) \in S_{21}, \quad (7.31) $$

with $x_d = x^{**}$ in the case (B5), and $x_d < x^{**}$ in the case (B6). Hence, with the same arguments as in the proof of proposition 7.10, (it is sufficient to exchange the roles of $x$ and $y$), one gets that there exists $\delta_0 > 0$ such that for any $j \in \mathbb{Z}_+^2$, the functions $h_{1j}$ and $y \mapsto yh_{2j}(y) = H_j(0, y)$ and can be analytically continued to the set $B(0, y_d + \delta_0) \setminus \{y_d\}$, and

$$ \begin{aligned} \lim_{y \to y_d} (y_d - y) y h_{2j}(y) &= \lim_{y \to y_d} (y_d - y) H_j(0, y) \\ &= c_1 \left( L_j(X_1(y_d), y_d) + (\phi_1(X_1(y_d), y_d) - 1) H_j(X_1(y_d), 0) \right) \end{aligned} \quad (7.32) $$
with

$$c_1 = \left. \left( \frac{d}{dy} \phi_2(X_1(y), y) \right)^{-1} \right|_{y=y_d} > 0. \qquad (7.33)$$

Since by (7.31), $y_d < y_P^{**}$, without any restriction of generality, we will suppose throughout our proof that

$$y_d + \delta_0 < y_P^{**}. \qquad (7.34)$$

Moreover, by using Proposition 7.1 and the first assertion of Corollary 7.6 we can inject $y = Y_1(x)$ to the functional equation (2.30):

- by Proposition 7.1, for some $\varepsilon > 0$, the function $Y_1$ is already analytically continued to $\tilde{U}_\varepsilon$ and extended as a continuous function to the set $U_\varepsilon$,

- by Corollary 7.6, there is $\delta_1 \in ]0, \varepsilon[$, such that any $j \in \mathbb{Z}_+^2$, the functions $x \mapsto L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x))$ and $x \mapsto -\psi_1(x, Y_1(x))h_{1j}(x) = (1 - \phi_1(x, Y_1(x)))H_j(x, 0)$ are analytic in the set $C(x_d - \delta_1, x_d)$ and for any $x \in C(x_d - \delta_1, x_d)$ the following relation holds

$$(1 - \phi_1(x, Y_1(x)))H_j(x, 0) = L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x)). \qquad (7.35)$$

To complete the proof of Lemma 7.5, the following steps will be performed:

step 1: First we will show that for some $\delta_2 \in ]0, \delta_1[$, the function $x \mapsto L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x))$ is analytic in the set

$$\{x \in C(x_d - \delta_2, x_d + \delta_2) \cap U_\varepsilon : Y_1(x) \neq Y_1(x_d)\}. \qquad (7.36)$$

step 2: Next we will prove that for some $\delta_3 \in ]0, \delta_2[$, the point $x_d$ is an only zero of the function $x \mapsto Y_1(x) - Y_1(x_d)$ in $C(x_d - \delta_3, x_d + \delta_3) \cap U_\varepsilon$, and we will deduce from our previous result that the function $x \mapsto L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x))$ is analytic in the set

$$C(x_d - \delta_3, x_d + \delta_3) \cap U_\varepsilon \setminus \{x_d\}. \qquad (7.37)$$

With this result and using the identity (7.35) we will be able to complete our proof.

Step 1: Since because of (7.31), $Y_1(x_d) = y_d < y_d + \delta_0$, by Proposition 7.4 applied with $\hat{y} = y_d + \delta_0$, we obtain that for some $0 < \delta_2 < \delta_1$, the function $x \mapsto (x, Y_1(x))$ maps the set $C(x_d - \delta_2, x_d + \delta_2) \cap U_\varepsilon$ to the set $\{(x, y) \in \mathcal{V} : |y| < y_d + \delta_0\}$. Since the function $y \mapsto H_j(0, y)$ is already analytically continued to the set $B(0, y_d + \delta_0) \setminus \{y_d\}$ and the functions $(x, y) \mapsto \psi_2(x, y)$ and $(x, y) \mapsto L_j(x, y)$ are analytic in $\mathcal{V}$, it follows that the function $x \mapsto L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x))$ is analytic in the set (7.36).

Step 2: by (7.31) and (2.9), we have

$$y_P^* \le y^* < y_d = y^{**} = Y_1(x_d),$$

and consequently, $(x_d, y_d) \neq (X_1(y_P^*), y_P^*)$. Since by (7.31), $(x_d, y_d) \in S_{21}$, using the definition of the curve $S_{21}$ (see (2.17)) one gets therefore

$$X_1(y_P^*) < x_d \le x_P^{**},$$

and consequently, using the assertions (ii) and (iv) of Proposition 7.1, we conclude that for some $0 < \delta_3 < \delta_2$, the point $x_d$ is an only zero of the function $x \mapsto Y_1(x) - Y_1(x_d)$ in $C(x_d - \delta_3, x_d + \delta_3) \cap U_\varepsilon$. Since with our previous result, we have already proved that the function $x \mapsto L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x))$ is analytic in the set (7.36), this proves that the function $x \mapsto L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x))$ is analytic in the set (7.37). Finally, since on the annulus $C(x_d - \delta_1, x_d)$, the identity (7.35) holds, this proves that the function $x \mapsto -\psi_1(x, Y_1(x))h_{1j}(x) = (1 - \phi_1(x, Y_1(x)))H_j(x, 0)$ can be analytically continued to the set $(C(x_d - \delta_3, x_d + \delta_3) \cap U_\varepsilon) \setminus \{x_d\}$. $\square$
### 7.6.2 Proof of Proposition 7.12

To complete the proof of Proposition 7.12, we consider separately all possible cases:

- when (B6) holds;

- when (B5) holds with $x_d = x^{**} < x_P^{**}$;

- when (B5) holds with $x_d = x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$;

- when (B5) holds with $x_d = x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) = 1$.

Suppose first that (B6) holds. Then by Proposition 2.2 and the definition of $x_d$ (see (2.33)), the following relations hold

$$x_P^* \le x^* < X_1(x^{**}) < x_d = X_2(y^{**}) < x^{**} \le x_P^{**}. \quad (7.38)$$

Hence, by Lemma 7.5 and according to the definition of the set $U_\epsilon$ (see (7.2)), for some $\hat{\delta} > 0$, the function $x \mapsto -\psi_1(x, Y_1(x))h_{1j}(x) = (1 - \phi_1(x, Y_1(x)))H_j(x, 0)$ is already analytically continued to the set $C(x_d - \hat{\delta}, x_d + \hat{\delta}) \setminus \{x_d\}$. By Proposition 7.7, for any $x \in C(x^*, x^{**})$,

$$|\phi_1(x, Y_1(x))| \le \phi_1(|x|, Y_1(|x|)) < 1.$$

Since by (7.38), $x^* < x_d < x^{**}$, it follows that for $0 < \delta' < \min\{x^{**} - x_d, x_d - x^*\}$, the function $x \mapsto 1/(1 - \phi_1(x, Y_1(x)))$ is analytic in $C(x_d - \delta', x_d + \delta')$ and

$$1 - \phi_1(x_d, Y_1(x_d)) > 0. \quad (7.39)$$

Since the function $x \mapsto \eta_j(x) = -\psi_1(x, Y_1(x))h_{1j}(x) = (1 - \phi_1(x, Y_1(x)))H_j(x, 0)$ is already extended as an analytic function to the set $C(x_d - \hat{\delta}, x_d + \hat{\delta}) \setminus \{x_d\}$, it follows that for $\delta = \min\{\delta', \hat{\delta}\}$, the function the function $x \mapsto H_j(x, 0)$ can be analytically continued to $C(x_d - \delta, x_d + \delta) \setminus \{x_d\}$ by letting

$$H_j(x, 0) = (1 - \phi_1(x, Y_1(x)))^{-1} (L_j(x, Y_1(x)) + \psi_2(x, Y_1(x))h_{2j}(Y_1(x))). \quad (7.40)$$

Remark finally that by (7.31), $y_d = y^{**} < y_P^{**}$. Hence, using exactly the same arguments as in the proof of Proposition 7.7 (it is sufficient to exchange the roles of $x$ and $y$), one gets $\phi_2(X_1(y_d), y_d) = \phi_2(X_1(y^{**}), y^{**}) = 1$. Since by (7.31), $X_1(y_d) < X_2(y_d) = x_d$ and $Y_1(x_d) = y_d$, and since under our hypotheses (see Assumption (A3)(vi)) the real valued function $x \mapsto \phi_2(x, y_d)$ is strictly increasing, it follows that

$$\phi_2(x_d, Y_1(x_d)) = \phi_2(x_d, y_d) = \phi_2(X_2(y_d), y_d) > \phi_2(X_1(y_d), y_d) = 1. \quad (7.41)$$

Using these relations together with (7.40), (7.39), (7.32) and (7.5), we obtain

$$\begin{align*}
\lim_{x \to x_d} (x_d - x)H_j(x, 0) &= \lim_{x \to x_d} (x_d - x)(1 - \phi_1(x, Y_1(x)))^{-1}\psi_2(x, Y_1(x))h_{2j}(Y_1(x)) \\
&= \lim_{x \to x_d} (x_d - x)(1 - \phi_1(x, Y_1(x)))^{-1}(\phi_2(x, Y_1(x)) - 1)H_j(0, Y_1(x)) \\
&= a_5\left(L_j(X_1(y_d), y_d) + (\phi_1(X_1(y_d), y_d) - 1)H_j(X_1(y_d), 0)\right) = a_5\varkappa_2(j)
\end{align*}$$

with

$$a_5 = (\phi_2(x, y) - 1) \left. \left( (1 - \phi_1(x, y)) \frac{d}{dy} \phi_2(X_1(y), y) \frac{d}{dx} Y_1(x) \right) \right|_{(x,y)=(x_d,y_d)}^{-1} > 0.$$

The first assertion of Proposition 7.12 is therefore proved.
Suppose now that (B5) holds and let $x_d = x^{**} < x_P^{**}$. Then by Proposition 2.2 and the definition of $x_d$ and $y_d$ (see (2.33) and (2.34)), one has

$$y_d = y^{**} = Y_1(x^{**}) = Y_1(x_d) \quad \text{and} \quad x^* < X_1(y_d) < x_d = X_2(y_d) = x^{**} < x_P^{**}.$$

and by Proposition 7.7, for some $\delta > 0$, the function $x \mapsto 1/(1 - \phi_1(x, Y_1(x)))$ is meromorphic in the annulus $C(x^*, x_d + \delta)$, and has there a unique and simple pole at the point $x_d = x^{**}$ with

$$\lim_{x \to x_d} \frac{x_d - x}{1 - \phi_1(x, Y_1(x))} = \left( \left. \frac{d}{dx} \phi_1(x, Y_1(x)) \right|_{x=x_d} \right)^{-1} > 0; \quad (7.42)$$

Here, the only difference with the previous case is that the function $x \mapsto 1/(1 - \phi_1(x, Y_1(x)))$ has a simple pole at $x_d = x^{**}$. Using therefore exactly the same arguments as in the previous case we obtain that for some $\delta > 0$, the function $x \mapsto H_j(x, 0)$ can be extended as an analytic function to the set $C(x_d - \delta, x_d + \delta)\setminus\{x_d\}$. And using finally (7.40) together with (7.32), (7.41), (7.5) and (7.42) we get

$$\lim_{x \to x_d} (x_d - x)^2 H_j(x, 0) = a_3(L_j(X_1(y_d), y_d) + (\phi_1(X_1(y_d), y_d) - 1)H_j(X_1(y_d), 0)) = a_3 \varkappa_2(j)$$

with

$$\begin{aligned} a_3 &= (\phi_2(x, Y_1(x)) - 1) \left( \left. \frac{d}{dx} \phi_1(x, Y_1(x)) \frac{d}{dy} \phi_2(X_1(y), y) \frac{d}{dx} Y_1(x) \right|_{x=x_d, y=y_d} \right)^{-1} \\ &= (\phi_2(x, y) - 1) \left. \left( \left. \frac{d}{dx} \phi_1(x, Y_1(x)) \frac{d}{dy} \phi_2(X_1(y), y) \frac{d}{dx} Y_1(x) \right|_{x=x_d, y=y_d} \right)^{-1} \right|_{x=x_d, y=y_d} > 0. \end{aligned}$$

The second assertion of Proposition 7.12 is therefore also proved.

Suppose now that (B5), $x_d = x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$ hold. Then by Lemma 7.5 and according to the definition of the set $U_\epsilon$ (see (7.2)), by using the identity (7.35), the function $x \mapsto -\psi_1(x, Y_1(x))h_{1j}(x)$ was already analytically continued to the set $C(x_P^{**} - \delta, x_P^{**} + \delta)\setminus[x_P^{**}, x_P^{**} + \delta]$, and by Proposition 7.7, for some $0 < \delta < \tilde{\delta}$, the function $x \mapsto (1 - \phi_1(x, Y_1(x)))^{-1}$ is analytic in $C(x_P^{**} - \delta, x_P^{**} + \delta)\setminus[x_P^{**}, x_P^{**} + \delta]$ and

$$H_j(x, 0) = xh_{1j}(x) = x\eta_j(x)(\psi_1(x, Y_1(x)))^{-1} = \eta_j(x)(1 - \phi_1(x, Y_1(x)))^{-1}, \quad \forall x \in C(x_P^{**} - \delta, x_P^{**}).$$

Hence, by using the identity (7.40), the function $x \mapsto H_j(x, 0)$ can be analytically continued to the set $C(x_P^{**} - \delta, x_P^{**} + \delta)\setminus[x_P^{**}, x_P^{**}]$, and using finally (7.40) together with (7.32), (7.6) and (7.31) one gets

$$\begin{aligned} & \lim_{x \to x_P^{**}} \sqrt{x_P^{**} - x} H_j(x, 0) \\ &= \lim_{x \to x_P^{**}} \sqrt{x_P^{**} - x} (1 - \phi_1(x, Y_1(x)))^{-1} (L_j(x, Y_1(x)) + \psi_2(x, Y_1(x)) h_{2j}(Y_1(x))) \\ &= \lim_{x \to x_P^{**}} \sqrt{x_P^{**} - x} (1 - \phi_1(x, Y_1(x)))^{-1} \psi_2(x, Y_1(x)) h_{2j}(Y_1(x)) \\ &= \lim_{x \to x_P^{**}} \sqrt{x_P^{**} - x} (1 - \phi_1(x, Y_1(x)))^{-1} (\phi_2(x, Y_1(x)) - 1) H_j(0, Y_1(x)) \\ &= \tilde{a}_4 (L_j(X_1(y_d), y_d) + (\phi_1(X_1(y_d), y_d) - 1)H_j(X_1(y_d), 0)) = \tilde{a}_4 \varkappa_2(j) \end{aligned}$$

with

$$\tilde{a}_4 = (\phi_2(x_d, y_d) - 1) \sqrt{\frac{\partial^2 P(x_d, y_d)}{\partial x P(x_d, y_d)}} \left( (1 - \phi_1(x_d, y_d)) \frac{d}{dy} \phi_2(X_1(y_d), y_d) \right)^{-1} > 0,$$