Consider now the case when (B5), $x_d = x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) = 1$ hold. In this case, with exactly the same arguments as above one gets that for some $\delta > 0$, the function $x \mapsto H_j(x, 0)$ can be continued as an analytic function to the set $C(x_P^{**} - \delta, x_P^{**} + \delta) \setminus [x_P^{**}, x_P^{**}]$ by using the identity (7.40). The only difference is here that now, by Proposition 7.7,

$$ \frac{1}{1 - \phi_1(x, Y_1(x))} \sim \frac{c_1}{\sqrt{x_P^{**} - x}} $$

with

$$ c_1 = \left. \left( \partial_y \phi_1(x, y) \sqrt{\partial_x P(x, y) / \partial_{yy}^2 P(x, y)} \right)^{-1} \right|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))} > 0 $$

and consequently, since in this case $x_d = x_P^{**}$ and $y_d = Y_1(x_P^{**})$, using (7.40) together with (7.32) and (7.31), one gets

$$ \lim_{x \to x_d} (x_d - x) H_j(x, 0) = a_4 \left( L_j(X_1(y_d), y_d) + (\phi_1(X_1(y_d), y_d) - 1) H_j(X_1(y_d), 0) \right) $$

with

$$ a_4 = \left. \left( \phi_2(x, y) - 1 \right) \partial_{yy}^2 P(x, y) \left( \partial_y \phi_1(x, y) \partial_x P(x, y) \frac{d}{dy} \phi_2(X_1(y), y) \right)^{-1} \right|_{(x,y)=(x_d, y_d)} > 0. $$

Proposition 7.12 is therefore proved.

## 7.7 Analytic continuation of the function $(x, y) \mapsto (1 - P(x, y))H_j(x, y)$

The following Lemma is the first step in the proof of the last assertion of Theorem 2.3.

**Lemma 7.6.** If the conditions (A1)-(A3) are satisfied and (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 $\mathbb{Z}_+^2$ and for any $j \in \mathbb{Z}_+^2 \setminus E_0$,

$$ \lim_{\substack{(x,y) \to (\hat{x}, \hat{y}) \\ (x,y) \in \mathring{D}}} (1 - P(x, y))(H_j(x, y) - H_j(x, 0) - H_j(0, y)) = \varkappa_{(\hat{x}, \hat{y})} \quad (7.43) $$

*Proof.* Indeed, by Theorem 2.1, for any $j \in \mathbb{Z}_+^2$ and $(\hat{x}, \hat{y}) \in S_{22}$ with $\hat{x} < x_d$ and $\hat{y} < y_d$, there is a neighborhood $\mathcal{V}(\hat{x}, \hat{y})$ of the point $(\hat{x}, \hat{y})$ in $\mathbb{R}^2$, such that for any $j \in \mathbb{Z}_+^2$, the function

$$ \begin{aligned} (x, y) &\mapsto L_j(x, y) + (\phi_1(x, y) - 1)H_j(x, 0) + (\phi_2(x, y) - 1)H_j(0, y) \\ &= L_j(x, y) + \psi_1(x, y)h_{1j}(x) + \psi_2(x, y)h_{2j}(y) \end{aligned} $$

is analytic in the polycircular set $\{(x, y) \in \mathbb{C}^2 : (|x|, |y|) \in \mathcal{V}(\hat{x}, \hat{y})\}$, and the function $(x, y) \mapsto (1 - P(x, y))(H_j(x, y) - H_j(x, 0) - H_j(0, y)) = Q(x, y)h_j(x, y)$ can be continued as an analytic function to $\{(x, y) \in \mathbb{C}^2 : (|x|, |y|) \in \mathcal{V}(\hat{x}, \hat{y})\}$ by letting

$$ (1 - P(x, y))(H_j(x, y) - H_j(x, 0) - H_j(0, y)) = L_j(x, y) + (\phi_1(x, y) - 1)H_j(x, 0) + (\phi_2(x, y) - 1)H_j(0, y). $$
Hence, for any $j \in \mathbb{Z}_+^2$ the quantity

$$ \kappa_{(\hat{x}, \hat{y})} = L_j(\hat{x}, \hat{y}) + (\phi_1(\hat{x}, \hat{y}) - 1)H_j(\hat{x}, 0) + (\phi_2(\hat{x}, \hat{y}) - 1)H_j(0, \hat{y}) $$

is well defined and as $(x, y) \to (\hat{x}, \hat{y})$ for $(x, y) \in \mathring{D}$, (7.43) holds. Moreover, since by Theorem 2.1, the function $(x, y) \mapsto H_j(x, y)$ is analytic in the set $\Omega_d(\Gamma) = \{(x, y) \in \mathbb{C}^2 : (|x|, |y|) \in \Gamma, |x| < x_d, |y| < y_d\}$ and since the set $\Omega_d(\Gamma)$ is a union of the poly-discs centered at the origin in $\mathbb{C}^2$, the power series

$$ H_j(x, y) - H_j(x, 0) - H_j(0, y) = \sum_{k_1=1}^{\infty} \sum_{k_2=1}^{\infty} g(j, k) x^{k_1} y^{k_2} $$

converge on the set $\Omega_d(\Gamma)$, and consequently, for $(x, y) \in \Gamma$ such that $x < x_d$ and $y < y_d$,

$$ H_j(x, y) - H_j(x, 0) - H_j(0, y) \geq 0. $$

Since for $(x, y) \in \mathring{D}$, $P(x, y) < 1$, it follows that for any $j \in \mathbb{Z}_+^2$, the left hand side of (7.43) is non-negative and consequently, the function $\kappa_{(\hat{x}, \hat{y})}$ is non-negative on $\mathbb{Z}_+^2$. $\square$

## 7.8 Harmonic functions $\kappa_i$, $\tilde{\kappa}_i$ and $\kappa_{(x,y)}$ and their properties

In this section, the second step of the proof of Theorem 2.3 is performed (see Section 7.1). This is a subject of the following statement.

**Proposition 7.13.** Under the hypotheses (A1) - (A3), the following assertions hold:

i) The function $\kappa_1$ defined by (2.35) is non-negative on $\mathbb{Z}_+^2$, positive on $\mathbb{Z}_+^2 \setminus E_0$ and harmonic for $(Z_{\tau_0}(n))$ in each of the following cases :

- if one of the assertions (B0), (B1), (B3) or (B4) is valid;

- if (B2) holds and $x^* < x_P^{**}$;

- if (B2) holds, $x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) = 1$.

ii) The function $\kappa_2$ defined by (2.37) is non-negative on $\mathbb{Z}_+^2$, positive on $\mathbb{Z}_+^2 \setminus E_0$ and harmonic for $(Z_{\tau_0}(n))$ in each of the following cases :

- if one of the assertions (B0), (B1), (B5) or (B6) is valid;

- if (B2) holds and $y^{**} < y_P^{**}$;

- if (B2) holds, $y^{**} = y_P^{**}$ and $\phi_1(X_1(y_P^{**}), y_P^{**}) = 1$.

iii) If (B2) is valid with $x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$, then the function $\tilde{\kappa}_1$ defined by (2.36) is non-negative on $\mathbb{Z}_+^2$, positive on $\mathbb{Z}_+^2 \setminus E_0$ and harmonic for $(Z_{\tau_0}(n))$.

iv) If (B2) is valid, then for any $(\hat{x}, \hat{y}) \in S_{22}$ with $\tilde{x} < x_d$ and $\hat{y} < y_d$, the function $\kappa_{(\hat{x}, \hat{y})}$ defined by (2.39) is non-negative on $\mathbb{Z}_+^2$, positive on $\mathbb{Z}_+^2 \setminus E_0$ and harmonic for $(Z_{\tau_0}(n))$.

To get this result, we first show there that the left hand sides of (2.40), (2.42), (2.44), (2.46), (2.48), (2.50), (2.52) and (2.54) are non-negative. With these arguments we conclude that each of the functions defined by (2.35), (2.36), (2.37), (2.39) is non-negative on $\mathbb{Z}_+^2$. Next we show that each of these functions (in the corresponding cases i)-iv). of our proposition) is harmonic for $(Z_{\tau_0}(n))$, and by using suitable Lyapunov functions, we prove that each of them is not identically zero on the set $\{j \in \mathbb{Z}_+^2; \|j\| \geq N_0\}$. With this results and using Lemma 2.2 we will be able to show that each of the functions defined by (2.35), (2.36), (2.37), (2.39) (in the corresponding cases i)-iv). of our proposition) is positive throughout the set $\mathbb{Z}_+^2 \setminus E_0$.
### 7.8.1 Preliminary estimates.

We begin the proof of this proposition with the following preliminary result

**Lemma 7.7.** 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 constant $C > 0$ (depending on the points $(x_1, y_1) \in \mathring{D} \cap \mathring{D}_1$ and $(x_2, y_2) \in \mathring{D} \cap \mathring{D}_2$ but do not depending on $j \in \mathbb{Z}_+^2$),

$$ \mathbb{P}_j(\tau_0 < +\infty) \le C(x_1^{j_1} y_1^{j_2} + x_2^{j_1} y_2^{j_2}), \quad \forall j \in \mathbb{Z}_+^2 \quad (7.44) $$

and

$$ \sum_{k \in \mathbb{Z}_+^2} g(j, k) (x_1^{k_1} y_1^{k_2} + x_2^{k_1} y_2^{k_2}) \le C (x_1^{j_1} y_1^{j_2} + x_2^{j_1} y_2^{j_2}), \quad \forall j \in \mathbb{Z}_+^2. \quad (7.45) $$

*Proof.* Indeed, if 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 by Lemma 6.1, for some $0 < \theta < 1$ and some finite subset $E$ of $\mathbb{Z}_+^2$ such that $(0, 0) \in E$ and the function $f: \mathbb{Z}_+^2 \to \mathbb{R}_+$ defined by $f(j_1, j_2)=x_1^{j_1} y_1^{j_2}+x_2^{j_1} y_2^{j_2}$, for all $(j_1, j_2) \in \mathbb{Z}_+^2$ satisfies the inequality

$$ \mathbb{E}_j(f(Z(1)); \tau_E > 1) \le \theta f(j), \quad \forall j = (j_1, j_2) \in \mathbb{Z}_+^2, $$

with $\tau_E = \inf\{n > 0 : Z(n) \in E\}$. By Lemma 6.2, it follows that for any $j \in \mathbb{Z}_+^2$, the series

$$ \sum_{k \in \mathbb{Z}^2} g(j, k) f(k) = \sum_{k \in \mathbb{Z}^2} g(j, k) (x_1^{k_1} y_1^{k_2} + x_2^{k_1} y_2^{k_2}) \quad (7.46) $$

converges, and moreover, (see (6.12) in the proof of Lemma 6.2), for

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

the following relation holds:

$$ \sum_{k \in \mathbb{Z}_+^2} g_E(j, k) f(k) \le \frac{f(j)}{1-\theta} = \frac{x_1^{j_1} y_1^{j_2} + x_2^{j_1} y_2^{j_2}}{1-\theta}. \quad (7.47) $$

Since $(0, 0) \in E$, from the last relation it follows in particular that for any $j \in \mathbb{Z}_+^2 \setminus E$,

$$ \begin{aligned} \mathbb{P}_j(\tau_0 < +\infty) \le \mathbb{P}_j(\tau_E < +\infty) &\le \left( \min_{\ell \in E} f(\ell) \right)^{-1} \sum_{k \in \mathbb{Z}_+^2} g_E(j, k) f(k) \\ &\le \left( \min_{\ell \in E} f(\ell) \right)^{-1} \frac{1}{1-\theta} (x_1^{j_1} y_1^{j_2} + x_2^{j_1} y_2^{j_2}) \end{aligned} \quad (7.48) $$

and consequently (7.44) is proved.

Remark finally that for any $j = (j_1, 0) \in \mathbb{Z}_+^2 \setminus E$,

$$ \begin{aligned} \sum_{k \in \mathbb{Z}_+^2} g(j, k) f(k) &= \sum_{k \in \mathbb{Z}_+^2} g_E(j, k) f(k) + \sum_{\ell \in E \setminus \{(0,0)\}} g_E(j, \ell) \sum_{k \in \mathbb{Z}_+^2} g(\ell, k) f(k), \\ &\le \sum_{k \in \mathbb{Z}_+^2} g_E(j, k) f(k) + \mathbb{P}_j(\tau_E < +\infty) \max_{\ell \in E} \sum_{k \in \mathbb{Z}_+^2} g(\ell, k) f(k). \end{aligned} $$

Since the set $E$ is finite and for any $j \in \mathbb{Z}_+^2$, the series (7.46) converge, this last relation combined with (7.47) and (7.48) prove (7.45). $\square$
**Lemma 7.8.** Under the hypotheses (A1)-(A3), for any $0 < y < y_d$ 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 for some $C > 0$,

$$H_j(0, y) \le C (x_1^{k_1} y_1^{k_2} + x_2^{k_1} y_2^{k_2}), \quad \forall j \in \mathbb{Z}_+^2. \qquad (7.49)$$

*Proof.* Indeed, if $0 < y < y_d$, then, by the definition of the set $\Theta$, the point $(0, Y_1(x^{**}))$ is in $\Theta$ and, consequently, by Lemma 6.4 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 relations (6.6) and (6.9) with $(x, y) = (0, Y_1(x^{**}))$. By Lemma 7.7, for these two points $(x_1, y_1)$ and $(x_2, y_2)$, one gets (7.45) with $(x, y) = (0, y)$, and consequently (7.49) holds. $\square$

### 7.8.2 Proof of the first two assertions of Proposition 7.13

Consider first the case when one of the following assertions holds:

- one of the cases (B0), (B1), (B3), (B4) holds;

- (B2) and $x_d < x_P^{**}$ hold;

- (B2), $x_d = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) = 1$ hold.

By Proposition 7.7 and the definition of $x_d$, we have always $x_d = x^{**}$ and

$$\phi_1(x_d, Y_1(x_d)) = 1. \qquad (7.50)$$

To prove Proposition 7.13 in each of the above cases, we first show that the function $\varkappa_1$ is non-negative and harmonic for $(Z_{\tau_0}(n))$. Next we will prove that $\varkappa_1(j_1, 0) > 0$ for all $j_1 > 0$ large enough, and using finally Lemma 2.2, we will conclude that the function $\varkappa_1$ is strictly positive on the set $\mathbb{Z}_+^2 \setminus E_0$.

Suppose first that either, one of the cases (B0), (B1), (B3), (B4) holds, or (B2) and $x_d < x_P^{**}$ hold. Then, by Proposition 7.10, for any $j \in \mathbb{Z}_+^2$,

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

with $c > 0$. Since $(x_d - x)H_j(x, 0) \ge 0$ for any real $x \in ]0, x_d[$, it follows that $\varkappa_1(j) \ge 0$ for any $j \in \mathbb{Z}_+^2$. When either one of the cases (B0), (B1), (B3), (B4) occurs or (B2) and $x^{**} < x_P^{**}$ hold, the function $\varkappa_1$ is therefore non-negative on $\mathbb{Z}_+^2$.

Suppose now that (B2), $x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) = 1$ hold. In this case, by Proposition 7.11, for any $j \in \mathbb{Z}_+^2$,

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

with $c > 0$. Since $\sqrt{x_d - x} H_j(x, 0) \ge 0$ for any real $x \in ]0, x_d[$, it follows that $\varkappa_1(j) \ge 0$ for any $j \in \mathbb{Z}_+^2$. Hence, in this case, the function $\varkappa_1$ is therefore also non-negative on $\mathbb{Z}_+^2$.

Harmonicity property of the function $\varkappa_1$ for the killed random walk. See Definition 1. For $j = (j_1, j_2) \in \mathbb{Z}_+^2$, by the definition of the functions $(x, y) \mapsto L_j(x, y)$ and $\varkappa_1 : \mathbb{Z}_+^2 \to \mathbb{R}_+$, we have

$$\varkappa_1(j) = \begin{cases} x_d^{j_1} (Y_1(x_d))^{j_2} - \mathbb{P}_j(\tau < +\infty) + (\phi_2(x_d, Y_1(x_d)) - 1)H_j(0, Y_1(x_d)) & \text{if } j \ne (0, 0), \\ \phi_0(x_d, Y_1(x_d)) - \mathbb{P}_{(0,0)}(\tau < +\infty) + (\phi_2(x_d, Y_1(x_d)) - 1)H_{(0,0)}(0, Y_1(x_d)) & \text{if } j = (0, 0). \end{cases}$$

Remark that for any $j = (j_1, j_2) \in \mathbb{Z}_+^2$,

$$\mathbb{E}_j (\mathbb{P}_{Z(1)}(\tau < +\infty); \tau > 1) = \mathbb{P}_j(\tau < +\infty) - \mu_1(-1, 0) \mathbb{1}_{\{(0,1)\}}(j_1, j_2) - \mu_2(0, -1) \mathbb{1}_{\{(1,0)\}}(j_1, j_2) \\ - \mu(-1, -1) \mathbb{1}_{\{(1,1)\}}(j_1, j_2) - \mu_0(0, 0) \mathbb{1}_{\{(0,0)\}}(j_1, j_2)$$
and since the function $j \mapsto H_j(0, Y_1(x_d))$ is potential for $(Z_{\tau_0}(n))$ (see the properties of potential for the Markov chains functions in [30]),

$$ \mathbb{E}_j(H_{Z(1)}(0, Y_1(x_d)), \tau > 1) = H_j(0, Y_1(x_d)) - (Y_1(x_d))^{j_2} \mathbb{1}_{\{j_1=0, j_2>0\}}(j_1, j_2), $$

where for $j, k \in \mathbb{Z}^2$ we denote

$$ \mathbb{1}_{\{k\}}(j) = \begin{cases} 1 & \text{if } j = k, \\ 0 & \text{otherwise,} \end{cases} $$

By the definition of the function $Y_1$, we have $P(x_d, Y_1(x_d)) = 1$, and because of (7.50), $\phi_1(x_d, Y_1(x_d)) = 1$. Hence, for any $j \in \mathbb{Z}_+^2$,

$$ \begin{aligned} \mathbb{E}_j(x_d^{Z_1(1)}(Y_1(x_d))^{Z_2(1)}; \tau > 1) &= \begin{cases} \phi_0(x_d, Y_1(x_d)) - \mu_0(0,0)\mathbb{1}_{\{(0,0)\}}(j_1, j_2) & \text{if } j = (0,0), \\ x_d^{j_1} \phi_1(x_d, Y_1(x_d)) - \mu_1(-1,0)\mathbb{1}_{\{(0,1)\}}(j_1, j_2) & \text{if } j_1 > 0 \text{ and } j_2 = 0, \\ (Y_1(x_d))^{j_2} \phi_2(x_d, Y_1(x_d)) - \mu_2(0,-1)\mathbb{1}_{\{(1,0)\}}(j_1, j_2) & \text{if } j_1 = 0 \text{ and } j_2 > 0, \\ x_d^{j_1} (Y_1(x_d))^{j_2} P(x_d, Y_1(x_d)) - \mu(-1,-1)\mathbb{1}_{\{(1,1)\}}(j_1, j_2) & \text{if } j_1 > 0, j_2 > 0, \end{cases} \\ &= \begin{cases} \phi_0(x_d, Y_1(x_d)) - \mu_0(0,0)\mathbb{1}_{\{(0,0)\}}(j_1, j_2) & \text{if } j = (0,0), \\ x_d^{j_1} - \mu_1(-1,0)\mathbb{1}_{\{(0,1)\}}(j_1, j_2) & \text{if } j_1 > 0 \text{ and } j_2 = 0, \\ (Y_1(x_d))^{j_2} \phi_2(x_d, Y_1(x_d)) - \mu_2(0,-1)\mathbb{1}_{\{(1,0)\}}(j_1, j_2) & \text{if } j_1 = 0 \text{ and } j_2 > 0, \\ x_d^{j_1} (Y_1(x_d))^{j_2} - \mu(-1,-1)\mathbb{1}_{\{(1,1)\}}(j_1, j_2) & \text{if } j_1 > 0, j_2 > 0, \end{cases} \end{aligned} $$

With these relations, for any $j = (j_1, j_2) \in \mathbb{Z}_+^2$ with $j_1 > 0$, we get

$$ \begin{aligned} \mathbb{E}_j(\varkappa_1(Z(1)), \tau > 1) &= x_d^{j_1} (Y_1(x_d))^{j_2} - \mathbb{P}_j(\tau < +\infty) + (\phi_2(x_d, Y_1(x_d)) - 1)H_j(0, Y_1(x_d)), \\ &= \varkappa_1(j), \end{aligned} \quad (7.51) $$

for any $j = (j_1, j_2) \in \mathbb{Z}_+^2$ with $j_1 = 0$ and $j_2 > 0$, we get

$$ \begin{aligned} \mathbb{E}_j(\varkappa_1(Z(1))) &= x_d^{j_1} (Y_1(x_d))^{j_2} \phi_2(x_d, Y_1(x_d)) - \mathbb{P}_j(\tau < +\infty) \\ &\quad + (\phi_2(x_d, Y_1(x_d)) - 1)(-x_d^{j_1} (Y_1(x_d))^{j_2} + H_j(0, Y_1(x_d))) \\ &= \varkappa_1(j). \end{aligned} \quad (7.52) $$

and for $j = (0, 0)$, we obtain

$$ \begin{aligned} \mathbb{E}_j(\varkappa_1(Z(1))) &= \phi_0(x_d, Y_1(x_d)) - \mathbb{P}_{(0,0)}(\tau < +\infty) + (\phi_2(x_d, Y_1(x_d)) - 1)H_{(0,0)}(0, Y_1(x_d)) \\ &= \varkappa_1(0,0). \end{aligned} \quad (7.53) $$

Relations (7.51), (7.52) and (7.53) prove that function $\varkappa_1$ is harmonic for $(Z_{\tau_0}(n))$.

To show that the function $\varkappa_1$ is strictly positive at some point $j = (j_1, j_2) \in \mathbb{Z}_+^2$, we investigate an asymptotic behavior of this function as $j_2 = 0$ and $j_1 \to +\infty$. By Proposition 2.2, in each of the cases (B0)-(B4), we have $0 < Y_1(x_d) < y_d$. Hence, by Lemma 7.8, 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) for which (7.49) holds for $y = Y_1(x_d)$ and consequently, by Lemma 7.7 we obtain (7.44). Using these relations for $j = (j_1, j_2) \in \mathbb{Z}_+^2$ with $j_2 = 0$ one gets

$$ |\varkappa_1(j_1, 0) - x_d^{j_1}| \le C_1 (x_1^{j_1} + x_2^{j_1}), \quad \forall j = (j_1, 0) \in \mathbb{Z}_+^2, \quad (7.54) $$

with some do not depending on $j$ constant $C_1 > 0$. Remark that $0 < x_2 < x_1$ because the points $(x_1, y_1)$ and $(x_2, y_2)$ satisfy (6.6), and that $x_1 < x_d$ because $(x_1, y_1) \in \mathring{D} \cap \mathring{D}_1$. Hence, from the last relation it follows that

$$ \varkappa_1(j_1, 0) \sim x_d^{j_1} \quad \text{as } j_1 \to \infty, $$
and consequently, there is $N_1 > 0$ such that

$$ \varkappa_1(j_1, 0) > 0 \quad \text{for any } j = (j_1, 0) \in \mathbb{Z}_+^2 \text{ with } j_1 \ge N_1. \qquad (7.55) $$

Now we are ready to complete the proof of the first assertion of Proposition 7.13: By Lemma 2.2, there are $N_0 > 0$ and a finite subset $E_0$ of $\mathbb{Z}_+^2$ such that $(0, 0) \notin E_0$ and for any $j \in \mathbb{Z}_+^2 \setminus E_0$ and $k \in \mathbb{Z}_+^2$ with $\|k\| > N_0$,

$$ g(j, k) = \sum_{n=0}^{\infty} \mathbb{P}_j(Z(n) = k, \tau_0 > n) > 0. $$

Hence, for any $j \in \mathbb{Z}_+^2 \setminus E_0$ and $k \in \mathbb{Z}_+^2$ with $\|k\| > N_0$, there is $n_{j,k} \in \mathbb{N}$ such that

$$ \mathbb{P}_j(Z(n_{j,k}) = k, \tau_0 > n_{j,k}) > 0. $$

For any $j \in \mathbb{Z}_+^2$, $k \in \mathbb{Z}_+^2$ and $n \in \mathbb{N}$, since the function $\varkappa_1$ is non-negative and harmonic for the killed random walk, one has

$$ \varkappa_1(j) = \mathbb{E}_j(\varkappa_1(Z(n)), \tau_0 > n) \ge \mathbb{P}_j(Z(n) = k, \tau_0 > n)\varkappa_1(k). $$

Using this relation together with (7.55), for $k = (k_1, 0) \in \mathbb{Z}_+^2$ with $k_1 \ge \max\{N_0, N_1\}$, one gets that for any $j \in \mathbb{Z}_+^2 \setminus E_0$

$$ \varkappa(j) \ge \mathbb{P}_j(Z(n_{j,k}) = k, \tau_0 > n_{j,k})\varkappa_1(k) > 0, $$

and consequently, the function $\varkappa_1$ is positive on $\mathbb{Z}_+^2 \setminus E_0$.

The first assertion of Proposition 7.13 is therefore proved. To get the second assertion of this proposition, it is sufficient to exchange the roles of $x$ and $y$.

### 7.8.3 Proof of the third assertion of Proposition 7.13

Suppose that (B2) holds and let $x^{**} = x_P^{**}$ and $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$. By the definition of the point $x_d$, we have here

$$ x_d = x^{**} = x_P^{**}, $$

and remark that in this case,

$$ \tilde{\varkappa}_1(j) = \partial_y \left( \frac{L_j(x, y) + (\phi_2(x, y) - 1) H_j(0, y)}{1 - \phi_1(x, y)} \right) \Big|_{(x,y)=(x_P^{**}, Y_1(x_P^{**}))}, \quad j \in \mathbb{Z}_+^2. $$

To show that the function $\tilde{\varkappa}_1$ is non-negative on $\mathbb{Z}_+^2$, we recall that in this cases, by Proposition 7.11,

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

with $c > 0$. Since for real $x \in ]0, x_P^{**}[$, the function $x \mapsto H_j(0, x)$ is increasing on $]0, x_P^{**}[$, it follows that $\tilde{\varkappa}_1(j) \ge 0$ for any $j \in \mathbb{Z}_+^2$.

Harmonicity property of the function $\tilde{\varkappa}_1$ for the killed random walk. For $j = (j_1, j_2) \in \mathbb{Z}_+^2$, we have

$$ \tilde{\varkappa}_1(j) = \frac{\partial_y \phi_1((x_P^{**}, Y_1(x_P^{**})))}{(1 - \phi_1(x_P^{**}, Y_1(x_P^{**})))^2} \varkappa_1(j) + \frac{1}{1 - \phi_1((x_P^{**}, Y_1(x_P^{**})))} \tilde{\varkappa}_1(j) \quad (7.56) $$
with $\tilde{\kappa}_1(j)$ defined by (2.35) and

$$ \tilde{\kappa}_1(j) = \partial_y L_j(x_P^{**}, Y_1(x_P^{**})) + (\phi_2(x_P^{**}, Y_1(x_P^{**})) - 1)\partial_y H_j(0, Y_1(x_P^{**})) \quad (7.57) \\ + \partial_y \phi_2((x_P^{**}, Y_1(x_P^{**})))H_j(0, Y_1(x_P^{**})), $$

where

$$ \partial_y L_j(x_P^{**}, Y_1(x_P^{**})) = \begin{cases} j_2 \times (x_P^{**})^{j_1} (Y_1(x_P^{**}))^{j_2-1} & \text{if } j = (j_1, j_2) \in \mathbb{Z}_+^2 \setminus \{(0,0)\}, \\ \partial_y \phi_0(x_P^{**}, Y_1(x_P^{**})) & \text{if } j = (j_1, j_2) = (0,0). \end{cases} $$

Remark now that for any $j = (j_1, j_2) \in \mathbb{Z}_+^2$, with the same arguments as in the proof of the first assertion of our proposition, one gets

$$ \mathbb{E}_j(\tilde{\kappa}_1(Z(1)), \tau_0 > 1) = \tilde{\kappa}_1(j) + (\phi_1(x_P^{**}, Y_1(x_P^{**})) - 1)(x_P^{**})^{j_1} \mathbb{1}_{\{j_1>0, j_2=0\}}(j_1, j_2) \quad (7.58) $$

(here, an only difference with the proof of the first assertion of our proposition is that $\phi_1(x_P^{**}, Y_1(x_P^{**})) < 1$). Moreover, since the functions $j \mapsto H_j(0, Y_1(x_P^{**}))$ and $j \mapsto \partial_y H_j(0, Y_1(x_P^{**}))$ are potential for $(Z_{\tau_0}(n))$ (see the properties of potential functions for Markov chains in Woess [30]), then for any $j = (j_1, j_2) \in \mathbb{Z}_+^2$, one has

$$ \mathbb{E}_j(H_{Z(1)}(0, Y_1(x_P^{**})), \tau_0 > 1) = H_j(0, Y_1(x_P^{**})) - (Y_1(x_P^{**}))^{j_2} \mathbb{1}_{\{j_1=0, j_2>0\}}(j_1, j_2) \quad (7.59) $$

and

$$ \mathbb{E}_j(\partial_y H_{Z(1)}(0, Y_1(x_P^{**})), \tau_0 > 1) = \partial_y H_j(0, Y_1(x_P^{**})) - j_2 \times (Y_1(x_P^{**}))^{j_2-1} \mathbb{1}_{\{j_1=0, j_2>0\}}. \quad (7.60) $$

Finally, straightforward calculation shows that

$$ \begin{aligned} \mathbb{E}_j(\partial_y L_{Z(1)}(x_P^{**}, Y_1(x_P^{**})), \tau_0 > 1) &= \mathbb{E}_j(Z_2(1)(x_P^{**})^{Z_1(1)}(Y_1(x_P^{**}))^{Z_2(1)-1}, \tau_0 > 1) \\ &= \partial_y \phi_1(x_P^{**}, Y_1(x_P^{**}))(x_P^{**})^{j_1} \mathbb{1}_{\{j_1>0, j_2=0\}}(j_1, j_2) \\ &+ (Y_1(x_P^{**}) \partial_y P(x_P^{**}, Y_1(x_P^{**}))) + j_2 P(x_P^{**}, Y_1(x_P^{**})) (j_2(x_P^{**})^{j_1} (Y_1(x_P^{**}))^{j_2-1} \\ &+ j_2 (Y_1(x_P^{**}))^{j_2-1} (Y_1(x_P^{**}) \partial_y \phi_2(x_P^{**}, Y_1(x_P^{**}))) + j_2 \phi_2(x_P^{**}, Y_1(x_P^{**})) \mathbb{1}_{\{j_1=0, j_1>0\}}(j_1, j_2). \end{aligned} $$

Since $\partial_y P(x_P^{**}, Y_1(x^{**})) = 0$ and $P(x_P^{**}, Y_1(x^{**})) = 1$, from the last relation it follows that

$$ \begin{aligned} \mathbb{E}_j(\partial_y L_{Z(1)}(x_P^{**}, Y_1(x_P^{**})), \tau_0 > 1) &= \partial_y \phi_1(x_P^{**}, Y_1(x_P^{**}))(x_P^{**})^{j_1} \mathbb{1}_{\{j_1>0, j_2=0\}}(j_1, j_2) + j_2(x_P^{**})^{j_1} (Y_1(x_P^{**}))^{j_2-1} \\ &+ j_2 (Y_1(x_P^{**}))^{j_2-1} (Y_1(x_P^{**}) \partial_y \phi_2(x_P^{**}, Y_1(x_P^{**}))) + j_2 \phi_2(x_P^{**}, Y_1(x_P^{**})) \mathbb{1}_{\{j_1=0, j_1>0\}}(j_1, j_2). \end{aligned} \quad (7.61) $$

When combined together, relations (7.56) - (7.60) imply that $\mathbb{E}_j(\tilde{\kappa}_1(Z(1), \tau_0 > 1) = \tilde{\kappa}_1(j)$ for any $j \in \mathbb{Z}_+^2$, and hence, the function $\tilde{\kappa}_1$ is harmonic for $(Z_{\tau_0}(n))$.

To show that the function $\tilde{\kappa}_1$ is strictly positive on $\mathbb{Z}_+^2 \setminus \{(0,0)\}$, we investigate an asymptotical behavior of this function as $j_2 = 0$ and $j_1 \to +\infty$. Remark that for any $j = (j_1, j_2) \in \mathbb{Z}_+^2 \setminus \{(0,0)\}$ with $j_2 = 0$,

$$ \tilde{\kappa}_1(j) = C_1(x_P^{**})^{j_1} + C_2 \mathbb{P}_j(\tau_0 < +\infty) + C_3 H_j(0, Y_1(x_P^{**})) + C_4 \partial_y H_j(0, Y_1(x_P^{**})) $$

with

$$ C_1 = \frac{\partial_y \phi_1((x_P^{**}, Y_1(x_P^{**})))}{(1 - \phi_1(x_P^{**}, Y_1(x_P^{**})))^2} \quad (7.62) $$

and some $C_2, C_3, C_4 \in \mathbb{R}$ do not depending on $j_1$. When (B2) and $x^{**} = x_P^{**}$ hold, we have

$$ x_d = x^{**} = x_P^{**}, \quad \text{and} \quad Y_1(x_P^{**}) < y_d = y^{**} \le y_P^{**}, $$
Hence by Lemma 7.8, for $\varepsilon > 0$ such that $Y_1(x^{**}) + \varepsilon < y_d$, 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) for which (7.49) holds with $y = Y_1(x^{**}) + \varepsilon$, and by Lemma 7.7 we get (7.44). Using these relations for $j = (j_1, j_2) \in \mathbb{Z}_+^2$ with $j_2 = 0$ we obtain

$$ \mathbb{P}_{(j_1,0)}(\tau_0 < +\infty) \le C_5 \frac{x_1^{j_1} + x_2^{j_1}}{1-\theta}, \quad \forall j = (j_1, 0) \in \mathbb{Z}_+^2, \qquad (7.63) $$

and

$$ H_{(j_1,0)}(0, Y_1(x_P^{**}) + \varepsilon) \le C_6(x_1^{j_1} + x_2^{j_1}) \qquad (7.64) $$

with some $C_5 > 0$ and $C_6 > 0$ do not depending on $j_1$. Since the function $y \mapsto H_j(0, y)$ is equal on the disk of its analyticity $B(0, y_d)$ to its power series with the positive coefficients, we have moreover

$$ \partial_y H_j(0, Y_1(x_P^{**})) \le C_7 H_j(0, Y_1(x_P^{**}) + \varepsilon_2) $$

with some do not depending on $j_1$ constant $C_7 > 0$, and consequently, using (7.62), (7.63) and (7.64), we obtain

$$ |\tilde{\kappa}_1(j_1, 0) - C_1(x_P^{**})^{j_1}| \le C_9(x_1^{j_1} + x_2^{j_1}), \quad \forall j_1 > 0, $$

with some do not depending on $j_1$ constant $C_8 > 0$. Remark finally that the constant $C_1$ defined by (7.62) is strictly positive because the function $y \mapsto \phi_1(x_P^{**}, y)$ is convex and strictly increasing on $]0, +\infty[$. Hence, with the same argument as in the proof of the first assertion of Proposition 7.13 we obtain

$$ \tilde{\kappa}_1(j_1, 0) \sim C_1(x_P^{**})^{j_1} \quad \text{as} \quad j_1 \to +\infty, $$

and consequently, $\tilde{\kappa}_1(j) > 0$ for all $j_1 > 0$ large enough.

The function $\tilde{\kappa}_1$ is therefore non-negative, harmonic for $(Z_{\tau_0}(n))$ and non zero at the points $(j_1, 0)$ for all $j_1 > 0$ large enough. With this results and using exactly the same arguments as in the proof of the first assertion of our proposition, we conclude that the function $\tilde{\kappa}_1$ is positive everywhere in $\mathbb{Z}_+^2 \setminus E_0$.

### 7.8.4 Proof of the last assertions of Proposition 7.13

Suppose now that (B2) holds, and let a point $(\hat{x}, \hat{y}) \in S_{22}$ be such that $\hat{x} < x_d$ and $\hat{y} < y_d$. For such a point $(\hat{x}, \hat{y})$, by Lemma 7.6, the function $\kappa_{(\hat{x}, \hat{y})}$ is non-negative on $\mathbb{Z}_+^2$.

The proof of the identity

$$ \mathbb{E}_j(\kappa_{(\hat{x}, \hat{y})}(Z(1)), \tau_0 > 1) = \kappa_{(\hat{x}, \hat{y})}(j), \quad \forall j \in \mathbb{Z}_+^2, $$

is straightforward, it uses the arguments quite similar to those of the proofs of (7.51), (7.52) and (7.53). The function $\kappa_{(\hat{x}, \hat{y})}$ is therefore harmonic for the killed random walk $(Z_{\tau_0}(n))$.

To prove that $\kappa_{(\hat{x}, \hat{y})}(j) > 0$ for some $j \in \mathbb{Z}_+^2$ we consider the point $\hat{w} = w_D(\hat{x}, \hat{y}) = (u_D(\hat{x}, \hat{y}), v_D(\hat{x}, \hat{y}))$ on the unit circle $S^1$ defined by (2.66) with $x = \hat{x}$ and $y = \hat{y}$, and we investigate an asymptotical behavior of $\kappa_{(\hat{x}, \hat{y})}(j)$ as $\|j\| \to +\infty$ and $j/\|j\| \to \hat{w}$.

By the definitions of the functions $L_j$ and $\kappa_{(\hat{x}, \hat{y})}$, for $j = (j_1, j_2) \in \mathbb{Z}_+^2 \setminus \{(0, 0)\}$,

$$ \kappa_{(\hat{x}, \hat{y})}(j) = \hat{x}^{j_1} \hat{y}^{j_2} - \mathbb{P}_j(\tau_0 < +\infty) + (\phi_1(\hat{x}, \hat{y}) - 1) H_j(\hat{x}, 0) + (\phi_2(\hat{x}, \hat{y}) - 1) H_j(0, \hat{y}) $$

Since we assume that $\hat{y} < y_d$, by Lemma 7.8, 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) for which (7.49) holds with $y = \hat{y}$, and by Lemma 7.7 we get
(7.44). With the same arguments (it is sufficient to exchange the roles of x and y, we get that there are two $(x_3, y_3) \in \mathring{D} \cap \mathring{D}_1$ and $(x_4, y_4) \in \mathring{D} \cap \mathring{D}_2$ for which

$$H_j(\hat{x}, 0) \le C' (x_3^{k_1} y_3^{k_2} + x_4^{k_1} y_4^{k_2}), \quad \forall j \in \mathbb{Z}_+^2,$$

with some do not depending on $j \in \mathbb{Z}_+^2$ constant $C' > 0$. Using the last relation together with (7.44) and (7.49), one gets

$$|\varkappa_{(\hat{x}, \hat{y})}(j) - \hat{x}^{j_1} \hat{y}^{j_2}| \le C_1 \sum_{i=1}^{4} x_i^{j_1} y_i^{j_2}. \qquad (7.65)$$

with some do not depending on $j$ constant $C_1 > 0$. For $\hat{w} = w_D(\hat{x}, \hat{y}) = (u_D(\hat{x}, \hat{y}), v_D(\hat{x}, \hat{y}))$ defined by relation (2.66) with $x = \hat{x}$ and $y = \hat{y}$, the point $(\hat{x}, \hat{y})$ is an only point in the set $D$ where the function $(x, y) \mapsto x^{\hat{u}_1} y^{\hat{u}_2}$ achieves its maximum over $D$. Since non of the points $(x_1, y_1), \dots, (x_4, y_4)$ is equal to $(\hat{x}, \hat{y})$ (recall that the points $(x_1, y_1), \dots, (x_4, y_4)$ belong to the interior of the set $D$ and the point $(\hat{x}, \hat{y})$ belongs to the boundary of $D$), it follows that

$$\frac{1}{\hat{x}^{j_1} \hat{y}^{j_2}} \sum_{i=1}^{4} x_i^{j_1} y_i^{j_2} \to 0 \quad \text{as} \quad \|j\| \to +\infty \quad \text{and} \quad j/\|j\| \to \hat{u}$$

and consequently, using (7.65), we get

$$\varkappa_{(\hat{x}, \hat{y})}(j) \sim \hat{x}^{j_1} \hat{y}^{j_2} \quad \text{as} \quad \|j\| \to +\infty \quad \text{and} \quad j/\|j\| \to \hat{u}.$$

This proves that $\varkappa_{(\hat{x}, \hat{y})}(j) > 0$ for any $j \in \mathbb{Z}_+^2$ with $\|j\|$ large enough and $j\|j\|$ closed enough to $\hat{u}$.

The function $\varkappa_{(\hat{x}, \hat{y})}$ is therefore non-negative, harmonic for $(Z_{\tau_0}(n))$, and non zero at some points $j = (j_1, j_2) \in \mathbb{Z}_+^2$ with $\|j\| > N_0$. Hence, the similar arguments as in the proof of the first assertion of our proposition, we conclude that the function $\tilde{\varkappa}_1$ is positive everywhere on $\mathbb{Z}_+^2 \setminus E_0$.

## 7.9 Proof of Theorem 2.3

Now we summarize the above results in orther to get Theorem 2.3:

- The first assertion of Theorem 2.3 follows from Proposition 7.10 and the first assertion of Proposition 7.13.

- The second assertion of Theorem 2.3 is a consequence of Proposition 7.11 and the third assertion of Proposition 7.13.

- Proposition 7.12, Proposition 7.13 and the second assertion of Proposition 7.13 prove the assertions iii)- v) of Theorem 2.3.

- The last assertion of our theorem is proved by Lemma 7.6 and the last assertion of Proposition 7.13.

# 8 Asymptotics along the Axes

This section is devoted to the proof of Theorem 2.4.

For $k_2=0$, the asymptotics (2.56)-(2.62) follows from Theorem 2.3 and the Tauberian-like theorem (see Corollary VI.1 of Flajolet and Sedgevick [10]) in a straightforward way. Hence,
to complete the proof of our theorem, i.e. to get the asymptotics (2.56)-(2.62) for $k_2 > 0$, it is
sufficient to show that for any $j \in \mathbb{Z}_+^2 \setminus E_0$,

$$
\lim_{n \to +\infty} \frac{g(j, (n, k_2))}{g(j, (n, 0))} = \nu_1(k_2) > 0, \quad \forall k_2 \in \mathbb{Z}^2. \tag{8.1}
$$

Before getting this result, let us notice that under the hypotheses (A1)-(A3), the function $y \mapsto \psi_1(x_d, y)/Q(x_d, y)$ is analytic at the origin $y=0$ because the functions $y \mapsto \psi_1(x_d, y)$ and $y \mapsto Q(x_d, y)$, see definitions (2.25) and (2.24), are analytic in a neighborhood of the closed disk $\bar{B}(0, Y_2(x_d))$ and with the definition of the function $Q$ (see (2.24))

$$
Q(x_d, 0) = - \sum_{\substack{k=(k_1, k_2) \in \mathbb{Z}^2 \\ k_2 = -1}} x_d^{k_1} \mu(k) < 0.
$$

For any given $n \in \mathbb{N}^*$, the quantity (2.55) is therefore well defined and equal to the $(n-1)$-th coefficient of the Taylor expansion of the function $y \mapsto \psi_1(x_d, y)/Q(x_d, y)$ in a neighborhood of the origin $y=0$:

$$
\psi_1(x_d, y)/Q(x_d, y) = \sum_{n=1}^{\infty} \nu_1(n)y^{n-1}.
$$

To show that all coefficients $\nu_1(n)$, $n \in \mathbb{N}$ are positive, and to get (8.1) for $k_2 > 0$, we will need
a probabilistic representation of the quantities $\nu_1(n)$, $n \in \mathbb{N}^*$, in terms of the invariant measure
of the following Markov chain on $\mathbb{Z}_+$: Define the twisted positive measures $\tilde{\mu}$ on $\mathbb{Z}^2$ and $\tilde{\mu}_1$ on
$\mathbb{Z} \times \mathbb{Z}_+$ by letting

$$
\begin{align*}
\tilde{\mu}(k_1, k_2) &= x_d^{k_1} (Y_1(x_d))^{k_2} \mu(k_1, k_2), \quad \forall (k_1, k_2) \in \mathbb{Z}^2, \\
\tilde{\mu}_1(k_1, k_2) &= x_d^{k_1} (Y_1(x_d))^{k_2} \mu_1(k_1, k_2), \quad \forall (k_1, k_2) \in \mathbb{Z} \times \mathbb{Z}_+.
\end{align*}
$$

With the definitions of the point $x_d$ and the function $Y_1$, we have

$$
\tilde{\mu}(\mathbb{Z}^2) = P(x_d, Y_1(x_d)) = 1 \quad \text{and} \quad \tilde{\mu}_1(\mathbb{Z} \times \mathbb{Z}_+) = \phi_1(x_d, Y_1(x_d)) \le 1,
$$

and consequently, the twisted measure $\tilde{\mu}$ is stochastic on $\mathbb{Z}_+^2$ and the twisted measure $\tilde{\mu}_1$ is
sub-stochastic on $\mathbb{Z} \times \mathbb{Z}_+$. Consider now a twisted random walk $(\tilde{Z}^1(n) = (\tilde{Z}_1^{(1)}(n), \tilde{Z}_2^{(1)}(n)))$ on
$\mathbb{Z} \times \mathbb{Z}_+$ with transition probabilities

$$
\mathbb{P}_{(j_1, j_2)}(\tilde{Z}^{(1)}(1) = (k_1, k_2)) = \begin{cases} \tilde{\mu}(k-j) & \text{if } j_2 > 0, \\ \tilde{\mu}_1(k-j) & \text{if } j_2 = 0. \end{cases}
$$

Since the transition probabilities of the twisted random walk $(\tilde{Z}^1(n) = (\tilde{Z}_1^{(1)}(n), \tilde{Z}_2^{(1)}(n)))$ are invariant with respect to the shifts on $(\ell, 0)$ for any $\ell \in \mathbb{Z}$, its second component $(\tilde{Z}_2^{(1)}(n))$ is a Markov chain on $\mathbb{Z}_+$ (stochastic if $\phi_1(x_d, Y_1(x_d)) = 1$ and sub-stochastic if $\phi_1(x_d, Y_1(x_d)) < 1$) with transition probabilities

$$
\mathbb{P}_{j_2}(\tilde{Z}_2^{(1)}(1) = k_2) = \begin{cases} \sum_{k_1=-1}^{\infty} \tilde{\mu}(k_1, k_2 - j_2) & \text{if } j_2 > 0, \\ \sum_{k_1=-1}^{\infty} \tilde{\mu}_1(k_1, k_2) & \text{if } j_2 = 0. \end{cases}
$$

Recall that a non-negative measure $\pi_1$ on $\mathbb{Z}_+$ is invariant for $(\tilde{Z}_2^{(1)}(n))$ if, for any $\ell \in \mathbb{Z}_+$, the relation

$$
\pi_1(\ell) = \sum_{\ell' \in \mathbb{Z}_+} \pi_1(\ell') \mathbb{P}_{\ell'}(\tilde{Z}_2^{(1)}(n) = \ell) \quad (8.2)
$$

holds. The following lemma relates the vector $\nu_1 = (\nu_1(k_2), k_2 \in \mathbb{Z}_+)$ with the unique (up to a multiplicative constant) invariant measure for $(\tilde{Z}_2^{(1)}(n))$.
**Lemma 8.1.** Under the hypotheses (A1)-(A3), $\pi_1 = (\pi_1(n) = \nu_1(n)(Y_1(x_d))^n, n \in \mathbb{N})$ is a unique up to the multiplication by constants invariant measure of the Markov chain $(\tilde{Z}_2^{(1)}(n))$.

*Proof.* We prove this lemma in two steps: First we will show that the Markov chain $(\tilde{Z}_2^{(1)}(n))$ has an invariant measure $\pi_1 = (\pi_1(n), n \in \mathbb{N})$ with $\pi_1(0) = 1$, and next we will prove that the generating function $f(y) = \sum_{n=0}^{\infty} \pi_1(n)(y/Y_1(x_d))^n$ satisfies in a neighborhood of the point $y = 0$ in $\mathbb{C}$ the identity

$$f(y) = \phi_1(x_d, y) + P(x_d, y)(f(y) - 1)$$

and we will deduce from this identity that $\pi_1(n) = \nu_1(n)(Y_1(x_d))^n$ for any $n \in \mathbb{N}$.

To perform the first step of our proof, let us notice that under our hypotheses, the jumps of the Markov chain $(\tilde{Z}_2^{(1)}(n))$ are integrable and moreover, for any non-zero $\ell \in \mathbb{Z}_+$,

$$\mathbb{E}_{\ell}(\tilde{Z}_2^{(1)}(1)) = \ell + \sum_{k_2=-1}^{\infty} k_2 \sum_{k_1=-1}^{\infty} x_d^{k_1} (Y_1(x_d))^{k_2} \mu(k_1, k_2) = \ell + Y_1(x_d) \partial_y P(x_d, Y_1(x_d))$$

with $\partial_y P(x_d, Y_1(x_d)) \le 0$. The Markov chain $(\tilde{Z}_2^{(1)}(n))$ is therefore recurrent if the measure $\tilde{\mu}_1$ is stochastic, i.e. if $\phi_1(x_d, Y_1(x_d)) = 1$, and it is transient if $\phi_1(x_d, Y_1(x_d)) < 1$.

In the case when the Markov chain $(\tilde{Z}_2^{(1)}(n))$ is recurrent, it has a unique invariant measure $(\pi_1(\ell), \ell \in \mathbb{Z}_+)$ with $\pi_1(0) = 1$ and

$$\pi_1(\ell) = \sum_{n=1}^{\infty} \mathbb{P}_0(\tilde{Z}_2^{(1)}(n) = \ell, \tilde{Z}_2^{(1)}(k) \neq 0, \forall k < n), \quad \forall \ell > 0.$$

Suppose now that the Markov chain $(\tilde{Z}_2^{(1)}(n))$ is transient, and let

$$\pi_1^{(k)}(\ell) = \sum_{n=0}^{\infty} \mathbb{P}_k(\tilde{Z}_2^{(1)}(n) = \ell), \quad k, \ell \in \mathbb{Z}_+.$$

Then for any $k, \ell \in \mathbb{Z}_+$ such that $k > \ell$, with a straightforward calculation one gets

$$\sum_{\ell' \in \mathbb{Z}_+} \pi_1^{(k)}(\ell') \mathbb{P}_{\ell'}(\tilde{Z}_2^{(1)}(1) = \ell) = \pi_1^{(k)}(\ell). \qquad (8.3)$$

Remark moreover that because of Assumption (A2), by the strong Markov property,

$$\pi_1^{(k)}(\ell)/\pi_1^{(k)}(0) = \pi_1^{(\ell)}(\ell)/\pi_1^{(\ell)}(0), \quad \forall k \ge \ell,$$

and $\mathbb{P}_{\ell'}(\tilde{Z}_2^{(1)}(1) = \ell) = 0$ for all $\ell' > \ell + 1$. Using (8.3), it follows that the measure $\pi_1 = (\pi_1(\ell), \ell \in \mathbb{Z}_+)$ with $\pi_1(0) = 1$ and

$$\pi_1(\ell) = \pi_1^{(\ell)}(\ell)/\pi_1^{(\ell)}(0), \quad \forall \ell \in \mathbb{N}^*,$$

is invariant for $(\tilde{Z}_2^{(1)}(n))$. The first step of our proof is therefore completed.

Remark now that for any $n \in \mathbb{Z}_+$, using the identity

$$\sum_{\ell'=0}^{\ell+n} \pi_1(\ell') \mathbb{P}_{\ell'}(\tilde{Z}_2^{(1)}(n) = \ell) = \pi_1(\ell) \qquad (8.4)$$

with $\ell = 0$, one gets

$$\pi_1(n) \le \pi_1(0) \left( \mathbb{P}_n(\tilde{Z}_2^{(1)}(n) = 0) \right)^{-1} \le \pi_1(0) \left( \sum_{k_1=-1}^{\infty} \tilde{\mu}(k_1, -1) \right)^{-n}$$
with

$$ \sum_{k_1=-1}^{\infty} \tilde{\mu}(k_1, -1) = \sum_{k_1=-1}^{\infty} x_d^{k_1} (Y_1(x_d))^{-1} \mu(k_1, -1) = -Q(x_d, 0) > 0. $$

Hence, the generating function $f(y) = \sum_{n=0}^{\infty} \pi_1(n)(y/Y_1(x_d))^n$ is analytic in a neighborhood of the origin $y = 0$ in $\mathbb{C}$, and by (8.2), for any non zero $y$, satisfies there the identity

$$ f(y) = f(0) \left( \sum_{(k_1, k_2) \in \mathbb{Z} \times \mathbb{Z}_+} \tilde{\mu}_1(k_1, k_2) (y/Y_1(x_d))^{k_2} \right) + (f(y) - f(0)) \left( \sum_{(k_1, k_2) \in \mathbb{Z} \times \mathbb{Z}} \tilde{\mu}(k_1, k_2) (y/Y_1(x_d))^{k_2} \right). $$

Since $f(0) = \pi_1(0) = 1$ and with the definition of the twisted measures $\tilde{\mu}$ and $\tilde{\mu}_1$, for non-zero $y$, we have

$$ \sum_{(k_1, k_2) \in \mathbb{Z} \times \mathbb{Z}} \tilde{\mu}(k_1, k_2) \frac{y^{k_2}}{(Y_1(x_d))^{k_2}} = P(x_d, y) $$

and

$$ \sum_{(k_1, k_2) \in \mathbb{Z} \times \mathbb{Z}_+} \tilde{\mu}_1(k_1, k_2) \frac{y^{k_2}}{(Y_1(x_d))^{k_2}} = \phi_1(x_d, y), $$

this implies that for any non-zero $y \in \mathbb{C}$ with $|y|$ small enough,

$$ (f(y) - 1)(1 - P(x_d, y)) = \phi_1(x_d, y) - 1. $$

By the definition of the functions $(x, y) \mapsto \psi_1(x, y)$ and $(x, y) \mapsto Q(x, y)$, from the last relation it follows that for any $y \in \mathbb{C}$ with $|y|$ small enough,

$$ (f(y) - 1)Q(x_d, y) = y\psi_1(x_d, y) $$

and consequently, since $Q(x_d, 0) \neq 0$, for any $y \in \mathbb{C}$ with $|y|$ small enough, we have also

$$ f(y) = 1 + \frac{y\psi_1(x_d, y)}{Q(x_d, y)}. $$

From the last relation, by the uniqueness of the coefficients of the Taylor expansion at the point $y = 0$, it follows that

$$ \pi_1(n)(Y_1(x_d))^{-n} = \nu_1(n), \quad \forall n \in \mathbb{Z}_+, $$

and consequently, the measure $\pi_1 = (\nu_1(n)Y_1(x_d))^n, n \in \mathbb{Z}_+$ is the unique invariant measure of the Markov chain $(\tilde{Z}_2^{(1)}(n))$ with $\pi_1(0) = 1$. $\square$

As a straightforward consequence of Lemma 8.1 one gets the following property of the coefficients $\nu_1(n), n \in \mathbb{N}$:

**Corollary 8.1.** Under the hypotheses (A1)-(A3), all coefficients $\nu_1(n), n \in \mathbb{Z}_+$, are positive and $(\nu_1(n), n \in \mathbb{Z}_+)$ is an only positive solution of the system

$$ \sum_{\ell_1=1}^{\infty} \mu_1(\ell_1, n) + \sum_{\ell_2=1}^{n+1} \nu_1(\ell_2) \sum_{\ell_1=1}^{\infty} \mu(\ell_1, n - \ell_2) = \nu_1(n), \quad k_2 \in \mathbb{Z}_+. \quad (8.5) $$

*Proof.* Indeed, since by Lemma 8.1, $\nu_1(n) = \pi_1(n)(Y_1(x_d))^{-n}$ for any $n \in \mathbb{Z}_+$, the system (8.5) is equivalent to (8.2). Under our hypotheses, the Markov chain $(\tilde{Z}_2^{(1)}(n))$ is irreducible on $\mathbb{Z}_+$ and $\pi_1(0) = 1 \neq 0$. It follows that $\pi_1(n) > 0$ for any $n \in \mathbb{Z}_+$, and consequently also $\nu_1(n) = \pi_1(n)(Y_1(x_d))^{-n} > 0$ for any $n \in \mathbb{Z}_+$. $\square$
From this result it follows that to get (8.1), it is sufficient to show that for any $j \in \mathbb{Z}_+^2 \setminus E_0$ and $k_2 \in \mathbb{N}^*$, the sequence $(g(j, (n, k_2))/g(j, (n, 0)))$ converges and the limits $\lim_{n \to \infty} g(j, (n, k_2))/g(j, (n, 0))$, $k_2 \in \mathbb{N}$, satisfy the system of the equations (8.5). To prove the convergence of these sequences, the following preliminary results will be needed.

**Corollary 8.2.** Under the hypotheses (A1)-(A3), if one of the conditions (B0)-(B6) holds, then for any $j \in \mathbb{Z}_+^2$

$$ \lim_{n} g(j, (k_1 + n, 0))/g(j, (n, 0)) = 1, \quad \forall k_1 \in \mathbb{Z}_+, \qquad (8.6) $$

and there are constants $c_j > 0$ and $c'_j > 0$ such that for any $k_1 > 0$,

$$ c_j k_1^{\gamma} x_d^{-k_1} \le g(j, (k_1, 0)) \le c'_j k_1^{\gamma} x_d^{-k_1}, \qquad (8.7) $$

where

$$ \gamma = \begin{cases} -1/2 & \text{if } x_d = x_P^{**} \text{ and either (B2) holds with } \phi_1(x_d, Y_1(x_d)) = 1, \\ & \text{or (B5) holds with } \phi_1(x_d, Y_1(x_d)) < 1, \\ -3/2 & \text{if (B2) holds with } x_d = x_P^{**} \text{ and } \phi_1(x_d, Y_1(x_d)) < 1, \\ 1 & \text{if (B5) holds with } x_d < x_P^{**}, \\ 0 & \text{otherwise.} \end{cases} $$

*Proof.* This result is a straightforward consequence of the asymptotics (2.56)-(2.62) with $k_2 = 0$. These asymptotics follow from Theorem 2.3 and a Tauberian-like theorem (see Corollary VI.1 of Flajolet and Sedgevick [10]) in a straightforward way. $\square$

**Lemma 8.2.** Under the hypotheses (A1)-(A3), for any $j \in \mathbb{Z}_+^2$ and $k_2 \in \mathbb{Z}_+$ there are three constants $c_1 > 0$, $c_2 > 0$ and $N(k_2) > 0$, such that

$$ c_1 g(j, (k_1, 0)) \le g(j, (k_1, k_2)) \le c_2 g(j, (k_1, 0)), \qquad (8.8) $$

for any $k_1 \in \mathbb{Z}_+$ such that $k_1 \ge N(k_2)$.

*Proof.* Remark that for any $j, k = (k_1, k_2) \in \mathbb{Z}_+^2$ and $N \in \mathbb{N}$, by using the Markov property one gets

$$ \begin{aligned} g(j, (k_1, k_2)) &\ge \sum_{n=N}^{\infty} \mathbb{P}_j(Z(n) = (k_1, k_2), Z(N) = (k_1, 0), \tau_0 > n) \\ &\ge g(j, (k_1, 0)) \mathbb{P}_{(k_1, 0)}(Z(N) = (k_1, k_2), \tau_0 > 0). \end{aligned} \qquad (8.9) $$

Moreover, similar arguments as in the proof of Lemma 7.2 shows that for any $k_2 \in \mathbb{Z}_+$, there is $N \in \mathbb{N}^*$ and a sequence $(\ell_1^{(0)}, \ell_2^{(0)}), \dots, (\ell_1^{(N)}, \ell_2^{(N)}) \in \mathbb{Z}^2$ such that

$$ \ell^{(0)} + \dots + \ell^{(n)} \in \mathbb{Z} \times \mathbb{Z}_+, \quad \forall n \in \{0, \dots, N\}, $$

$$ \sum_{n=0}^{N} \ell^{(n)} = (0, k_2) \quad \text{and} \quad \mu_1(\ell^{(0)}) \prod_{n=1}^{N} \mu(\ell^{(n)}) > 0. $$

Hence, using Assumption (A2) we conclude that for any $(k_1, k_2) \in \mathbb{Z}_+^2$ with $k_1 > N$,

$$ \mathbb{P}_{(k_1, 0)}(Z(n(k_2)) = (k_1, k_2), \tau_0 > 0) \ge \mu_1(\ell^{(0)}) \prod_{n=1}^{N} \mu(\ell^{(n)}) > 0. $$
When combined with (8.9) the last relation proves the first inequality of (8.8) with

$$c_1 = \mu_1(\ell^{(0)}) \prod_{n=1}^{N} \mu(\ell^{(n)}).$$

The proof of the second inequality of (8.8) is quite similar.

<figure><img src="image_5.png" /></figure>

Now we are ready to complete proof of Theorem 2.4. Throughout our proof, the starting point $j \in \mathbb{Z}_+^2 \setminus E_0$ will be given.

Since by Lemma 8.2 for any $k_2 \in \mathbb{Z}_+$, the sequence $(g(j, (n, k_2))/g(j, (n, 0)), n \ge N_n)$ is bounded below and above by some positive constants, to get (8.1) it is sufficient to show that for any subsequence $(N_n)$ of the sequence of non-zero natural numbers $(n)$, for which the sequence of functions

$$k_2 \mapsto f_n(k_2) = g(j, (N_n, k_2))/g(j, (N_n, 0)) \quad (8.10)$$

converges point-wise in $\mathbb{Z}_+$, one has

$$\lim_{n} f_n(k_2) = \nu_1(k_2), \quad \forall k_2 \in \mathbb{Z}_+. \quad (8.11)$$

Suppose now that for a subsequence $(N_n)$, the sequence of functions $(f_n)$ defined by (8.10) converges point-wise in $\mathbb{Z}_+$ and let $f_\infty = \lim_n f_n$. By Corollary 8.1, to get (8.11) it is sufficient to show that the limit function $f_\infty$ satisfies the system of equations

$$\sum_{\ell_1=1}^{\infty} \mu_1(\ell_1, k_2) + \sum_{\ell_2=1}^{n+1} f_\infty(\ell_2) \sum_{\ell_1=1}^{\infty} \mu(\ell_1, k_2 - \ell_2) = f_\infty(k_2), \quad k_2 \in \mathbb{Z}_+. \quad (8.12)$$

To get this result, we consider the sequence of functions $F_n : \mathbb{Z} \times \mathbb{Z}_+ \to \mathbb{R}_+$ defined for any $(k_1, k_2) \in \mathbb{Z} \times \mathbb{Z}_+$ by

$$F_n(k_1, k_2) = \begin{cases} g(j, (k_1 + N_n, k_2))/g(j, (N_n, 0)) & \text{if } k_1 + N_n \ge 0, \\ 0 & \text{otherwise.} \end{cases}$$

Remark that by (8.6), the sequence of functions $(F_n)$ also converges point-wise in $\mathbb{Z} \times \mathbb{Z}_+$, and for any $(k_1, k_2) \in \mathbb{Z} \times \mathbb{Z}_+$,

$$\lim_{n} F_n(k_1, k_2) = \lim_{n} f_n(k_2) = f_\infty(k_2). \quad (8.13)$$

Remark moreover that for any $k = (k_1, k_2) \in \mathbb{Z}_+^2 \setminus \{j\}$,

$$\begin{aligned} g(j, (k_1, k_2)) &= \sum_{(\ell_1, \ell_2) \in \mathbb{Z}_+^2} g(j, (\ell_1, \ell_2)) \mathbb{P}_{(\ell_1, \ell_2)}(Z(1) = (k_1, k_2), \tau_0 > 1) \\ &= \sum_{\ell_2=1}^{k_2+1} g(j, (0, \ell_2)) \mu_2(k_1, k_2 - \ell_2) + \sum_{\ell_1=1}^{k_1+1} g(j, (\ell_1, 0)) \mu_1(k_1 - \ell_1, k_2) \\ &\quad + \sum_{\ell_1=1}^{k_1+1} \sum_{\ell_2=1}^{k_2+1} g(j, (\ell_1, \ell_2)) \mu(k_1 - \ell_1, k_2 - \ell_2). \end{aligned}$$

Using this relation with $k_1 = N_n$, for any $n \in \mathbb{N}$ such that $N_n > j_1$, one gets

$$F_n(0, k_2) = \sum_{\ell_2=1}^{k_2+1} F_n(-N_n, \ell_2) \mu_2(N_n, k_2 - \ell_2) + \sum_{\ell_1=-1}^{N_n} F_n(-\ell_1, 0) \mu_1(\ell_1, k_2) \\ + \sum_{\ell_1=-1}^{N_n} \sum_{\ell_2=1}^{k_2+1} F_n(-\ell_1, \ell_2) \mu(\ell_1, k_2 - \ell_2) \quad \forall k_2 \in \mathbb{Z}_+,$$
and consequently, using (8.13) we will obtain (8.12) if we prove that for any $k_2 \in \mathbb{Z}_+$ and $\ell_2 \in \{1, \dots, k_2 + 1\}$, the following relations hold

$$ \lim_{n} F_{n}(-N_{n}, \ell_{2})\mu_{2}(N_{n}, k_{2} - \ell_{2}) = 0, \qquad (8.14) $$

$$ \lim_{n} \sum_{\ell_1=-1}^{N_n} F_n(-\ell_1, 0)\mu_1(\ell_1, k_2) = \sum_{\ell_1=-1}^{\infty} \lim_{n} F_n(-\ell_1, 0)\mu_1(\ell_1, k_2) \qquad (8.15) $$

and

$$ \lim_{n} \sum_{\ell_1=-1}^{N_n} F_n(-\ell_1, \ell_2)\mu(\ell_1, k_2 - \ell_2) = \sum_{\ell_1=-1}^{\infty} \lim_{n} F_n(-\ell_1, \ell_2)\mu(\ell_1, k_2 - \ell_2). \qquad (8.16) $$

To get (8.14) we remark that by Corollary 8.2, there is a constant $C > 0$ such that for any $N_n$ large enough,

$$ F_n(-N_n, \ell_2) = g(j, (0, \ell_2))/g(j, (N_n, 0)) \le g(j, (0, \ell_2))C(N_n)^\gamma x_d^{N_n} $$

and consequently, for any $\varepsilon > 0$, there is a constant $C_\varepsilon > 0$ such that for any $N_n$ large enough,

$$ F_n(-N_n, \ell_2) \le g(j, (0, \ell_2))C_\varepsilon(1 + \varepsilon)^{N_n}x_d^{N_n}. $$

Since by Assumption (A3)(ii), the generating function $\phi_1$ is finite in a neighborhood of the point $(x_d, Y_1(x_d))$, there is $\varepsilon > 0$ such that for any $k_2 \in \mathbb{Z}_+$,

$$ \lim_{n \to \infty} \mu_1(n, k_2)(1 + \varepsilon)^n x_d^n = 0. $$

and consequently, for any $k_2 \in \mathbb{Z}_+$ and $\ell_2 \in \{1, \dots, k_2 + 1\}$, (8.14) holds.

To get (8.15) we use the implicit function theorem and Corollary 8.2. By Corollary 8.2, there is a constant $C_j > 0$ such that for any $N_n > 0$ and $\ell_1 \in \mathbb{Z}$ such that $-1 \le \ell_1 \le N_n$,

$$ F_n(-\ell_1, 0) = \frac{g(j, (N_n - \ell_1, 0))}{g(j, (N_n, 0))} \le C_1 \frac{(N_n - \ell_1)^\gamma}{N_n^\gamma} x_d^{\ell_1}. \qquad (8.17) $$

In the case when $\gamma \ge 0$, it follows that for any $N_n > 0$ and $\ell_1 \in \mathbb{Z}$ such that $-1 \le \ell_1 \le N_n$,

$$ F_n(-\ell_1, 0) \le C_1 x_d^{\ell_1}, $$

and consequently, since $F_n(-\ell_1, 0) = 0$ for all $\ell_1 \ge N_n$, and since under our hypotheses,

$$ \sum_{\ell_1=-1}^{\infty} x_d^{\ell_1} \mu_1(\ell_1, k_2) \le (Y_1(x_d))^{-k_2} \sum_{\ell=(\ell_1, \ell_2) \in \mathbb{Z} \times \mathbb{Z}_+} \sum_{\ell_1=1}^{\infty} x_d^{\ell_1} (Y_1(x_d))^{\ell_2} \mu_1(\ell_1, \ell_2) = \phi_1(x_d, Y_1(x_d)) < +\infty, $$

by the implicit function theorem one gets (8.15). In the case when $\gamma \ge 0$, (8.15) is therefore proved.

Suppose now that $\gamma < 0$. In this case, using (8.17), one gets that for any $N_n > 0$ and $\ell_1 \in \mathbb{Z}$ such that $-1 \le \ell_1 < N_n$,

$$ F_n(-\ell_1, 0) \le C_1 \frac{N_n^{\gamma}}{(N_n - \ell_1)^{|\gamma|}} x_d^{\ell_1} \le \begin{cases} C_1 N_n^{|\gamma|} x_d^{\ell_1} \le C_1 (2\ell_1)^{|\gamma|} x_d^{\ell_1} & \text{if } \ell_1 > N_n/2 \\ C_1 2^{|\gamma|} x_d^{\ell_1} & \text{if } \ell_1 \le N_n/2 \end{cases} $$

and consequently, for any $\varepsilon > 0$ there is a constant $C_\varepsilon > 0$ such that $N_n > 0$ and $\ell_1 \in \{-1, \dots, N_n - 1\}$, one has

$$ F_n(-\ell_1, 0) \le C_\varepsilon (1 + \varepsilon)^{\ell_1} x_d^{\ell_1}. \qquad (8.18) $$
Since $F_n(-\ell_1, 0) = 0$ for all $\ell_1 \ge N_n$, and since under our hypotheses, for some $\varepsilon > 0$ small enough,

$$ \sum_{\ell_1=-1}^{\infty} (1+\varepsilon)^{\ell_1} x_d^{\ell_1} \mu_1(\ell_1, k_2) \le (Y_1(x_d))^{-k_2} \sum_{\ell=(\ell_1, \ell_2) \in \mathbb{Z} \times \mathbb{Z}_+} (1+\varepsilon)^{\ell_1} x_d^{\ell_1} (Y_1(x_d))^{\ell_2} \mu_1(\ell_1, \ell_2) \\ \le (Y_1(x_d))^{-k_2} \phi_1((1+\varepsilon)x_d, Y_1(x_d)) < +\infty, \quad (8.19) $$

by the implicit function theorem, it follows (8.15). Relation (8.15) is therefore proved.

To get (8.16) we use first Lemma 8.2. By Lemma 8.2, there are two constants $N(\ell_2)$ and $C_1 > 0$ such that for any $N_n \ge 0$ and $\ell_1 \in \{-1, \dots, N_n - N(\ell_2)\}$,

$$ F_n(-\ell_1, \ell_2) = \frac{g(j, (N_n - \ell_1, \ell_2))}{g(j, (N_n, 0))} \le C_1 \frac{g(j, (N_n - \ell_1, 0))}{g(j, (N_n, 0))} $$

and for $\ell_1 \in \{N_n - N(\ell_2) + 1, \dots, N_n - 1\}$,

$$ F_n(-\ell_1, \ell_2) = \frac{g(j, (N_n - \ell_1, \ell_2))}{g(j, (N_n, 0))} \frac{C_2}{g(j, (N_n, 0))} $$

with $C_2 = \max\{g(j, (1, 0)), \dots, g(j, (N(\ell_2), 0))\}$. Hence with the similar arguments as above one gets that for any $\varepsilon > 0$ there is a constant $C_\varepsilon > 0$ such that $N_n > 0$ and $\ell_1 \in \{-1, \dots, N_n - 1\}$, (8.18) holds, and consequently, using (8.19) we conclude that (8.16) holds.

Theorem 2.4 is therefore proved.

# 9 Asymptotics along Directions of $S_+^1$

This section is devoted to the proof of Theorem 2.6.

## 9.1 Main ideas and the sketch of the proof

The main ideas of the proof of Theorem 2.6 are the following:

By using Theorem 2.1, we first get an integral representation

$$ g(j, k) = \int_{|x|=\hat{x}} \int_{|y|=\hat{y}} \frac{L_j(x, y) + (\phi_1(x, y) - P(x, y))H_j(x, 0) + (\phi_2(x, y) - P(x, y))H_j(0, y)}{(2\pi i)^2 x^{k_1+1} y^{k_2+1} (1-P(x, y))} dxdy \quad (9.1) $$

for any $(\hat{x}, \hat{y}) \in \Gamma$ with $\hat{x} < x_d$ and $\hat{y} < y_d$. Next we show that the set $\{(x, y) \in \mathring{D} : x < x_d, y < y_d\}$ is non empty. By the definition of the set $\Gamma$, the set $\mathring{D}$ is included to the set $\Gamma$, with this result we will be able to consider the integral representation (9.1) with $\hat{x} < x_d$, $\hat{y} < y_d$ such that $(\hat{x}, \hat{y}) \in \mathring{D}$.

To prove the first tree assertions of Theorem 2.6, the integral representation (9.1) is next modified in the following way: since the functions $(x, y) \mapsto L_j(x, y)$ and $(x, y) \mapsto (1-P(x, y))^{-1}$ are analytic in the polycircular set $\Omega(\mathring{D})$, and, by Theorem 2.1,

- the function $(x, y) \mapsto (\phi_1(x, y) - P(x, y))H_j(x, 0)$ is analytic in the polycircular set $\{(x, y) \in \Omega(\mathring{D}) : |x| < x_d\}$,

- the function $(x, y) \mapsto (\phi_2(x, y) - P(x, y))H_j(0, y)$ is analytic in the polycircular set $\{(x, y) \in \Omega(\mathring{D}) : |y| < y_d\}$,
and since the set $\Omega(\mathring{D})$ does not contain zeros of the function $(x, y) \mapsto x^{k_1+1}y^{k_2+1}(1-P(x, y))$, we can write

$$g(j, k) = I_0(j, k) + I_1(j, k) + I_2(j, k) \qquad (9.2)$$

with

$$I_0(j, k) = \frac{1}{(2\pi i)^2} \int_{|x|=\hat{x}_0} \int_{|y|=\hat{y}_0} \frac{L_j(x, y)}{x^{k_1+1}y^{k_2+1}(1-P(x, y))} dx dy, \qquad (9.3)$$

$$I_1(j, k) = \frac{1}{(2\pi i)^2} \int_{|x|=\hat{x}_1} \int_{|y|=\hat{y}_1} \frac{(\phi_1(x, y) - P(x, y))H_j(x, 0)}{x^{k_1+1}y^{k_2+1}(1-P(x, y))} dx dy \qquad (9.4)$$

and

$$I_2(j, k) = \frac{1}{(2\pi i)^2} \int_{|x|=\hat{x}_2} \int_{|y|=\hat{y}_2} \frac{(\phi_2(x, y) - P(x, y))H_j(0, y)}{x^{k_1+1}y^{k_2+1}(1-P(x, y))} dx dy \qquad (9.5)$$

for any $(\hat{x}_0, \hat{y}_0) \in \mathring{D}$, $(\hat{x}_1, \hat{y}_1) \in \{(x, y) \in \mathring{D} : x < x_d\}$, and $(\hat{x}_2, \hat{y}_2) \in \{(x, y) \in \mathring{D} : y < y_d\}$.

To prove the first assertion of Theorem 2.6, it is sufficient to get the asymptotic behavior (2.75) when $\min\{k_1, k_2\} \to +\infty$ and $w_k = k/\|k\| \to w$ for any $w \in W_1$. In order to get this result, we identify the asymptotic behavior of $I_1(j, k)$ by using the residue theorem (applied first for the integral with respect to $x$ and next for the integral with respect to $y$), and using next large deviation estimates of $I_0(j, k)$ and $I_2(j, k)$ we prove that the terms $I_0(j, k)$ and $I_3(j, k)$ are negligible with respect to $I_1(j, k)$.

The proof of the second assertion of Theorem 2.6 is exactly the same as the proof of the first assertion, it is sufficient to exchange the roles of the first and the second coordinates of the points $j, k \in \mathbb{Z}_+^2$.

To prove the assertions (iii)-(v) of our theorem, we show that the term $I_0(j, k)$ is negligible with respect to $I_1(j, k) + I_2(j, k)$ and we identify (in the same way as in the proof of the first assertion of our theorem) the asymptotic behavior of each term $I_1(j, k)$ and $I_2(j, k)$.

Finally, the last assertion of Theorem 2.6, is obtained as a consequence of Proposition 1 of the paper [18].

The proof of Theorem 2.6 is organized as follows:

The integral representations (9.1) and (9.2) of the Green function $g(j, k)$ are obtained respectively in Lemma 9.1 and Corollary 9.1 of Section 9.2.

Section 9.3 is devoted to the upper large deviation estimates for the integrals $I_0(j, k)$, $I_1(j, k)$ and $I_2(j, k)$. These large deviation estimates will be used in order to identify the dominant terms of the sum $I_0(j, k)+I_0(j, k)+I_0(j, k)$ of the right-hand side of (9.2).

In Section 9.4 we obtain exact asymptotics of $I_1(j, k)$ (resp $I_2(j, k)$) as $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w$ for those $w = (u, v) \in \mathbb{S}_+^1$ for which $u > u_D(x_d, Y_2(x_d))$ (resp. for which $v > v_D(X_2(y_d), y_d)$). This is a subject of Lemma 9.6 and Lemma 9.5 below. Remark that we do not need the exact asymptotics of $I_1(j, k)$ (resp $I_2(j, k)$) when $k/\|k\| \to w = (u, v) \in \mathbb{S}_+^1$ and $u \le u_D(x_d, Y_2(x_d))$ (resp. $v \le v_D(X_2(y_d), y_d)$): the large deviation asymptotics obtained in Section 9.3 will be in this case sufficient.

In Section 9.5 the proof of the first assertion of our theorem is completed. It will be proved there that for any $w \in W_1$, the inequality $u > u_D(x_d, Y_2(x_d))$ holds, and consequently that the results of Section 9.4 provide the exact asymptotic for $I_1(j, k)$ as $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w \in W_1$. A comparison of the exact asymptotics for $I_1(j, k)$ as $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w \in W_1$ with the asymptotics of $I_2(j, k)$ and $I_0(j, k)$ will show that the terms $I_0(j, k)$ and $I_2(j, k)$ are negligible with respect to $I_1(j, k)$.

The second assertion of Theorem 2.6 is obtained by using the arguments of the symmetry: to get this statement, the same arguments as in the proof of the first assertion can be applied if one exchanges the roles of $x$ and $y$.
In Section 9.6 and 9.7, we complete the proof of the assertions (iii)-(v) of Theorem 2.6. It will be shown in these cases, the exact asymptotics of $I_1(j, k)$ and $I_2(j, k)$ are given by the results of Section 9.4 and that the term $I_0(j, k)$ is negligible with respect to the terms $I_1(j, k)$ and $I_2(j, k)$. The exact asymptotic of the Green function $g(j, k)$ will be obtained by comparing the exact asymptotics of $I_1(j, k)$ and $I_2(j, k)$.

## 9.2 Integral representation of the coefficients $g(j, k)$

**Lemma 9.1.** Under the assumptions (A1)-(A4), the set $\{(x, y) \in \mathring{D} : x < x_d, y < y_d\}$ is non empty and for any $(\hat{x}, \hat{y}) \in \mathring{D}$ such that $\hat{x} < x_d$ and $\hat{y} < y_d$, (9.1) holds.

*Proof.* By the first assertion of Theorem 2.1, the series (2.28) (and consequently also the series (6.1)) converge on the polycircular set $\{(x, y) \in \Omega(\Gamma) : |x|<x_d, |y|<y_d\}$. The function $(x, y) \mapsto H_j(x, y)$ is therefore analytic in $\{(x, y) \in \Omega(\Gamma) : |x|<x_d, |y|<y_d\}$ and for any $(\hat{x}, \hat{y}) \in \Gamma$ with $\hat{x}<x_d$ and $\hat{y}<y_d$, one has

$$g(j, k) = \frac{1}{(2\pi i)^2} \int_{|x|=\hat{x}} \int_{|y|=\hat{y}} \frac{H_j(x, y)}{x^{k_1+1} y^{k_2+1}} dx dy, \quad \forall j, k \in \mathbb{Z}_+^2 \quad (9.6)$$

Remark now that under our assumptions, the set $\{(x, y) \in \mathring{D} : x < x_d, y < y_d\}$ is non empty:

- when one of the assertions (B0)-(B2) is valid, this is a consequence of the first assertion of Lemma 6.4

- when one of the assertions (B3) or (B4) holds, the set $\{(x, y) \in \mathring{D} : x < x_d, y < y_d\}$ is non empty because in this case and $x_P^* < x_d = x^{**} < x_P^{**}$, $y_d > Y_1(x_d)$ and by Lemma 4.1, any point of the line segment $[(x_d, Y_1(x_d)), (x_d, Y_2(x_d))]$, aside of the ends points $(x_d, Y_1(x_d))$ and $(x_d, Y_2(x_d))$ belongs to the interior $\mathring{D}$ of the set $D$;

- similarly, when one of the assertions (B5) or (B6) holds, the set $\{(x, y) \in \mathring{D} : x<x_d, y<y_d\}$ is non empty because $y_P^*<y_d=y^{**}<y_P^{**}$, $x_d > X_1(y_d)$ and by Lemma 4.1, any point of line segment $[(X_1(y_d), y_d), (X_2(y_d), y_d)]$, aside of the end points $(X_1(y_d), y_d)$ and $(X_2(y_d), y_d)$ belongs to the interior of the set $D$.

By the definition of the set $\Gamma$, the set $\mathring{D}$ is included to $\Gamma$, it follows that (9.6) holds also for any $(\hat{x}, \hat{y}) \in \mathring{D}$ with $\hat{x} < x_d$ and $\hat{y} < y_d$.

Furthermore, by the second assertion of Theorem 2.1, for any $j \in \mathbb{Z}_+^2$, on the set $\{(x, y) \in \Omega(\Gamma) : |x|<x_d, |y|<y_d\}$, the function $h_j$ satisfies the identity (2.30). Since the set $\{(x, y) \in \Omega(\mathring{D}) : x < x_d, y < y_d\}$ has an empty intersection with the sets $\{(x, y) \in \mathbb{C}^2 : x = 0\}$ and $\{(x, y) \in \mathbb{C}^2 : y = 0\}$, it follows that on the set $\{(x, y) \in \Omega(\mathring{D}) : x<x_d, y<y_d\}$, for any $j \in \mathbb{Z}_+^2$, the relation

$$(1 - P(x, y))(H_j(x, y) - H_j(x, 0) - H_j(0, y)) = Q(x, y)h_j(x, y) \\ = L_j(x, y) + \psi_1(x, y)h_{1j}(x) + \psi_2(x, y)h_{2j}(y) \\ = L_j(x, y) + (\phi_1(x, y) - 1)H_j(x, 0) + (\phi_2(x, y) - 1)H_j(0, y),$$

holds, or equivalently,

$$(1 - P(x, y))H_j(x, y) = L_j(x, y) + (\phi_1(x, y) - P(x, y))H_j(x, 0) + (\phi_2(x, y) - P(x, y))H_j(0, y).$$
Since the function $(x, y) \mapsto 1/(1 - P(x, y))$ is analytic in the set $\{(x, y) \in \mathbb{C}^2 : (|x|, |y|) \in \mathring{D}\}$, this implies that, on the set $\{(x, y) \in \mathbb{C}^2 : (|x|, |y|) \in \mathring{D}, |x| < x_d, |y| < y_d\}$, we have also the identity

$$H_j(x, y) = \frac{L_j(x, y) + (\phi_1(x, y) - P(x, y))H_j(x, 0) + (\phi_2(x, y) - P(x, y))H_j(0, y)}{1 - P(x, y)}.$$

The last identity combined with (9.6) proves (9.1). □

Since the functions $(x, y) \mapsto L_j(x, y)$ and $(x, y) \mapsto x^{-k_1-1}y^{k_2-1}(1-P(x, y))^{-1}$ are analytic in the polycircular set $\Omega(\mathring{D})$, and by Theorem 2.1, the functions $(x, y) \mapsto (\phi_1(x, y) - P(x, y))H_j(x, 0)$ and $(x, y) \mapsto (\phi_2(x, y) - P(x, y))H_j(0, y)$ are analytic respectively on $\{(x, y) \in \Omega(\mathring{D}) : |x|<x_d\}$ and $\{(x, y) \in \Omega(\mathring{D}) : |y|<y_d\}$, as a straightforward consequence of Lemma 9.1 we obtain

**Corollary 9.1.** Under the assumptions (A1)-(A4), for any $j \in \mathbb{Z}_+^2$ and $k=(k_1, k_2) \in \mathbb{Z}_+^2 \setminus \{(0, 0)\}$, relation (9.2) holds with $I_0(j, k)$, $I_1(j, k)$ and $I_2(j, k)$ defined respectively by (9.3), (9.4) and (9.5), for any $(\hat{x}_0, \hat{y}_0) \in \mathring{D}$, $(\hat{x}_1, \hat{y}_1) \in \{(x, y) \in \mathring{D} : x<x_d\}$, and $(\hat{x}_2, \hat{y}_2) \in \{(x, y) \in \mathring{D} : y<y_d\}$.

### 9.3 Large deviation estimates of the quantities $I_0(j, k)$, $I_1(j, k)$ and $I_2(j, k)$

**Lemma 9.2.** Under the assumptions (A1)-(A4), for any $w = (u, v) \in \mathbb{S}_+^1$, and $j, k \in \mathbb{Z}_+^2$, as $\|k\| \to \infty$ and $k/\|k\| \to w$,

$$\limsup_k \|k\|^{-1} \log |I_0(j, k)| \le - \max_{(\hat{x}, \hat{y}) \in D} (u \ln(\hat{x}) + v \ln(\hat{y})), \quad (9.7)$$

$$\limsup_k \|k\|^{-1} \log |I_1(j, k)| \le - \max_{(\hat{x}, \hat{y}) \in D, \hat{x} \le x_d} (u \ln(\hat{x}) + v \ln(\hat{y})). \quad (9.8)$$

and

$$\limsup_k \|k\|^{-1} \log |I_2(j, k)| \le - \max_{(\hat{x}, \hat{y}) \in D, \hat{y} \le y_d} (u \ln(\hat{x}) + v \ln(\hat{y})). \quad (9.9)$$

**Proof.** Indeed, the definition of $I_2(j, k)$ and Cauchy's inequality give, for any $j, k = (k_1, k_2) \in \mathbb{Z}_+^2$ and $(\hat{x}, \hat{y}) \in \mathring{D}$ with $\hat{y} < y_d$,

$$|I_2(j, k)| \le M(\hat{x}, \hat{y}) \hat{x}^{-k_1} \hat{y}^{-k_2},$$

with

$$M(\hat{x}, \hat{y}) = \max\{|(\phi_2(x, y) - P(x, y))H_j(0, y)(1 - P(x, y))^{-1}| : (x, y) \in \mathbb{C}^2, |x| = \hat{x}, |y| = \hat{y}\}.$$

This proves that as $\|k\| \to +\infty$ and $k/\|k\| \to w = (u, v)$,

$$\limsup_k \|k\|^{-1} \log |I_2(j, k)| \le - \sup_{(\hat{x}, \hat{y}) \in \mathring{D}: \hat{y} < y_d} (u \log(\hat{x}) + v \log(\hat{y})) \\ = - \max_{(\hat{x}, \hat{y}) \in D, \hat{y} \le y_d} (u \log(\hat{x}) + v \log(\hat{y}))$$

where the second relation holds because the function $(x, y) \mapsto u \ln x + v \ln y$ is continuous on $D$. Relation (9.9) is therefore proved. The proof of (9.7) and (9.8) is quite similar. □
## 9.4 Exact asymptotic behavior of $I_1(jk)$ as $\min\{k_1, k_2\} \to \infty$

To get the exact asymptotic of $I_1(jk)$ as $\min\{k_1, k_2\} \to \infty$ we consider first the following preliminary results.

**Lemma 9.3.** Under the assumption (A1), for any $w = (u, v) \in \mathbb{S}_+^1$, the function $x \mapsto u \ln(x) + v \ln(Y_2(x))$ is strictly increasing in the line segment $[x_P^*, x_D(w)]$ and strictly decreasing in the line segment $[x_D(w), x_P^{**}]$.

*Proof.* Indeed, recall that under our assumptions, the function $(\alpha, \beta) \mapsto \tilde{P}(\alpha, \beta) = P(e^{\alpha}, e^{\beta})$ is strictly convex and finite in a neighborhood of the set $\tilde{D} = \{(\alpha, \beta) \in \mathbb{R}^2 : P(e^{\alpha}, e^{\beta}) \le 1\}$, and that for any $w = (u, v) \in \mathbb{S}^1$, the point $(\alpha_D(w), \beta_D(w)) = (\ln(x_D(w)), \ln(y_D(w)))$ (see Definition 2.6.1) is an only point on the boundary of the set $\tilde{D}$, where the function $(\alpha, \beta) \mapsto u\alpha + v\beta$ achieves its maximum over $\tilde{D}$.

Consider the line segment $[\alpha_P^*, \alpha_P^{**}] = \{\alpha \in \mathbb{R} : \inf_{\beta \in \mathbb{R}} \tilde{P}(\alpha, \beta) \le 1\}$. We have, see the proof of Lemma 4.1, $\alpha_P^*=x \ln(x_P^*)$, $\alpha_P^{**}=\ln(x_P^{**})$, and that for any $\alpha \in [\alpha_P^*, \alpha_P^{**}]$ and $\beta_2(\alpha) = \ln(Y_2(e^{\alpha}))$, we have

$$ \tilde{P}(\alpha, \beta_2(\alpha)) = 1 \quad \text{and} \quad \partial_{\beta} \tilde{P}(\alpha, \beta_2(\alpha)) > 0. $$

Since the function $\tilde{P}$ is strictly convex, by the implicit function theorem, it follows that the function $\beta_2$ is also strictly convex on the line segment $[\alpha_P^*, \alpha_P^{**}]$.

Under our assumptions, the definition 2.6.1 of the mapping $w \mapsto (\alpha_D(w), \beta_D(w))=(\ln(x_D(w)), \ln(y_D(w)))$ gives that, if $w = (u, v) \in \mathbb{S}_+^1$,

$$ \partial_{\beta} \tilde{P}(\alpha_D(w), \beta_D(w)) \ge 0. $$

By the definition of the function $\alpha \mapsto \beta_2(\alpha) = \ln(Y_2(e^{\alpha}))$, see the proof of Lemma 4.1, it follows that,

$$ \beta_D(w) = \beta_2(\alpha_D(w)), $$

and that $\alpha_D(w)$ is an only point in the line segment $[\alpha_P^*, \alpha_P^{**}]$ where the function

$$ \alpha \mapsto u\alpha + v\beta_2(\alpha) = u\alpha + v \ln(Y_2(e^{\alpha})) \quad (9.10) $$

achieves its maximum over $[\alpha_P^*, \alpha_P^{**}]$. Since the function $\beta_2$ is strictly convex on $[\alpha_P^*, \alpha_P^{**}]$, the function (9.10) is also strictly convex on $[\alpha_P^*, \alpha_P^{**}]$, and consequently, it is strictly increasing on the segment $[\alpha_P^*, \alpha_D(w)]$ and strictly decreasing on the line segment $[\alpha_D(w), \alpha_P^{**}]$. Since the function $x \mapsto \ln(x)$ is strictly increasing on $]0, +\infty[$, this proves that the function $x \mapsto u \ln(x) + v \ln(Y_2(x))$ is strictly increasing on the line segment $[x_P^*, x_D(w)]$ and strictly decreasing on the line segment $[x_D(w), x_P^{**}]$. $\square$

**Lemma 9.4.** If condition (A1) is satisfied and let, for some $x_0 \in [x_P^*, x_P^{**}]$, $\varepsilon_1 > 0$ and $\varepsilon_2 > 0$, a function $(x, y) \mapsto F(x, y)$ be analytic in the polycircular set

$$ \{(x, y) \in \mathbb{C}^2 : |x_0 - |x|| < \varepsilon_1, |Y_2(x_0) - |y|| < \varepsilon_2\} $$

and do not vanish at the point $(x_0, Y_2(x_0))$. Then for any $w = (u, v) \in \mathbb{S}_+^1$ such that $x_D(w) > x_0$ and $(\hat{x}, \hat{y}) \in \tilde{D}$ such that $x_0 - \varepsilon_1 < \hat{x} < x_0$ and $|Y_2(x_0) - \hat{y}| < \varepsilon_2$, as $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w$,

$$ \frac{1}{(2\pi i)^2} \int_{|x|=\hat{x}} \int_{\|y\|=\hat{y}} \frac{F(x, y)}{x^{k_1} y^{k_2} (x_0 - x)(1 - P(x, y))} dx dy \sim \frac{C_1}{x_0^{k_1} (Y_2(x_0))^{k_2}} \quad (9.11) $$

with

$$ C_1 = \frac{F(x, y)}{\partial_y P(x, y)} \Big|_{(x,y)=(x_0, Y_2(x_0))} \neq 0 \quad (9.12) $$
and

$$ \frac{1}{(2\pi i)^2} \int_{|x|=\hat{x}} \int_{\|y\|=\hat{y}} \frac{F(x, y)}{x^{k_1} y^{k_2} (x_0 - x)^2 (1 - P(x, y))} dx dy \\ \sim \frac{k_1 C_1}{x_0^{k_1+1} (Y_2(x_0))^{k_2}} - \frac{k_2 C_2}{x_0^{k_1} (Y_2(x_0))^{k_2+1}} \quad (9.13) $$

with $C_1$ given by (9.12) and

$$ C_2 = \frac{F(x, y) \partial_x P(x, y)}{(\partial_y P(x, y))^2} \Bigg|_{(x,y)=(x_0, Y_2(x_0))} . \quad (9.14) $$

*Proof.* Before proving this lemma, remark that under our assumptions, by Lemma 4.1, $Y_2(x_0) > Y_1(x_0)$ and consequently, without any restriction of generality, we can assume throughout our proof that

$$ \varepsilon_2 < Y_2(x_0) - Y_1(x_0). $$

Because of Assumption (A1), the function $(x, y) \mapsto 1 - P(x, y)$ is analytic in a neighborhood $\mathcal{V}$ of the set $\Omega(D)$ and does not vanishes in its interior $\Omega(\mathring{D})$. Hence, for any $m \in \mathbb{N}^*$, the function

$$ (x, y) \mapsto F(x, y)(x_0 - x)^{-m} x^{-k_1} y^{-k_2} (1 - P(x, y))^{-1} $$

is analytic in the polycircular sets $\{(x, y) \in \Omega(\mathring{D}), x_0 - \varepsilon_1 < |x| < x_0, |Y_2(x_0) - y| < \varepsilon_2\}$ and $\{(x, y) \in \Omega(\mathring{D}), x_0 < |x| < x_0 + \varepsilon_1, |Y_2(x_0) - y| < \varepsilon_2\}$, and consequently, the function

$$ (\hat{x}, \hat{y}) \mapsto J_{m,k}(\hat{x}, \hat{y}) = \frac{1}{(2\pi i)^2} \int_{|x|=\hat{x}} \int_{\|y\|=\hat{y}} \frac{F(x, y) \, dx \, dy}{x^{k_1} y^{k_2} (x_0 - x)^m (1 - P(x, y))} $$

is constant on the set $A_- = \{(x, y) \in \mathring{D} : x_0 - \varepsilon_1 < x < x_0, |Y_2(x_0) - y| < \varepsilon_2\}$ and on the set $A_+ = \{(x, y) \in \mathring{D} : x_0 < x < x_0 + \varepsilon_1, |Y_2(x_0) - y| < \varepsilon_2\}$. We denote

$$ J_m(k) = J_{m,k}(\hat{x}, \hat{y}) \quad \text{for } (\hat{x}, \hat{y}) \in A_-, $$

$$ \hat{J}_m(k) = J_{m,k}(\hat{x}, \hat{y}) \quad \text{for } (\hat{x}, \hat{y}) \in A_+. $$

Remark now that for any $\hat{y} \in ]Y_2(x_0) - \varepsilon_2, Y_2(x_0)[$ and $\delta > 0$ small enough, the point $(x_0 - \delta, \hat{y})$ belongs to the set $A_-$, the point $(x_0 + \delta, \hat{y})$ belongs to the set $A_+$ and by the residue theorem, for any $y \in \mathbb{C}$ with $|y| = \hat{y}$,

$$ \int_{|x|=x_0+\delta} \frac{F(x, y) \, dx}{x^{k_1}(x_0 - x)(1 - P(x, y))} = \int_{|x|=x_0-\delta} \frac{F(x, y) \, dx}{x^{k_1}(x_0 - x)(1 - P(x, y))} - 2\pi i \frac{F(x_0, y)}{x_0^{k_1}(1 - P(x_0, y))} $$

and

$$ \int_{|x|=x_0+\delta} \frac{F(x, y) \, dx}{x^{k_1}(x_0 - x)^2(1 - P(x, y))} = \int_{|x|=x_0-\delta} \frac{F(x, y) \, dx}{x^{k_1}(x_0 - x)^2(1 - P(x, y))} \\ + 2\pi i \left( \frac{\partial_x F(x_0, y)}{x_0^{k_1}(1 - P(x_0, y))} - \frac{k_1 F(x_0, y)}{x_0^{k_1+1}(1 - P(x_0, y))} + \frac{F(x_0, y) \partial_x P(x_0, y)}{x_0^{k_1}(1 - P(x_0, y))^2} \right) $$

Hence, for any $\hat{y} \in ]Y_2(x_0) - \varepsilon_2, Y_2(x_0)[$,

$$ J_1(k) = \hat{J}_1(k) + \frac{1}{2\pi i x_0^{k_1}} \int_{\|y\|=\hat{y}} \frac{F(x_0, y)}{y^{k_2}(1 - P(x_0, y))} dy \quad (9.15) $$
and

$$J_2(k) = \hat{J}_2(k) - \frac{1}{2\pi i x_0^{k_1}} \int_{\|y\|=\hat{y}} \frac{\partial_x F(x_0, y)}{y^{k_2}(1-P(x_0, y))} dy \\ + \frac{k_1}{(2\pi i)x_0^{k_1+1}} \int_{\|y\|=\hat{y}} \frac{F(x_0, y)}{y^{k_2}(1-P(x_0, y))} dy - \frac{1}{2\pi i x_0^{k_1}} \int_{\|y\|=\hat{y}} \frac{F(x_0, y)\partial_x P(x_0, y)}{y^{k_2}(1-P(x_0, y))^2} dy. \quad (9.16)$$

Due to Assumption (A1)(ii), the function $y \mapsto P(x_0, y)$ is analytic in a neighborhood of the closed annulus $\{y \in \mathbb{C} : Y_1(x_0) \le |y| \le Y_2(x_0)\}$, satisfies the inequality

$$|P(x_0, y)| \le P(x_0, |y|) < 1$$

on the annulus $\{y \in \mathbb{C} : Y_1(x_0) < |y| < Y_2(x_0)\}$, and because of Assumption (A1) i) (we use here Proposition P7.5 of [29]), for any $y \in \mathbb{C}$ with $|y| = Y_2(x_0)$,

$$|P(x_0, y)| < P(x_0, Y_2(x_0)) = 1 \quad \text{whenever } y \ne Y_2(x_0).$$

Since by Lemma 4.1,

$$\partial_y P(x, y)|_{(x,y)=(x_0, Y_2(x_0))} > 0,$$

this proves that for some $\delta > 0$, the point $Y_2(x_0)$ is an only and simple zero of the function $y \mapsto P(x_0, y)$ in the annulus $\{y \in \mathbb{C} : Y_1(x_0) < |y| < Y_2(x_0) + \delta\}$. Since we assumed that $Y_2(x_0) - \varepsilon_2 < Y_1(x_0)$, and the functions $y \mapsto F(x_0, y)$, $y \mapsto \partial_x F(x_0, y)$, $y \mapsto F(x_0, y)$ are analytic in the annulus $\{y \in \mathbb{C} : Y_2(x_0) - \varepsilon_2 < |y| < Y_2(x_0) + \varepsilon_2\}$, this implies that for $\delta > 0$ small enough, the functions

$$y \mapsto \frac{F(x_0, y)}{y^{k_2}(1-P(x_0, y))} \quad \text{and} \quad y \mapsto \frac{\partial_x F(x_0, y)}{y^{k_2}(1-P(x_0, y))}$$

are analytic in the set $\{y \in \mathbb{C} : Y_2(x_0) - \varepsilon_2 < |y| < Y_2(x_0) + \delta, y \ne Y_2(x_0)\}$ and have at the point $Y_2(x_0)$, a simple pole with the residue equal respectively to

$$\frac{-C_1}{(Y_2(x_0))^{k_2}} \quad \text{and} \quad \frac{-\tilde{C}_1}{(Y_2(x_0))^{k_2}}.$$

with $C_1$ given by (9.12) and

$$\tilde{C}_1 = \left. \frac{\partial_x F(x, y)}{\partial_y P(x, y)} \right|_{(x,y)=(x_0, Y_2(x_0))}.$$

Similarly, the function

$$y \mapsto \frac{F(x_0, y) \partial_x P(x_0, y)}{y^{k_2}(1-P(x_0, y))^2}$$

is analytic in the set $\{y \in \mathbb{C} : Y_2(x_0) - \varepsilon_2 < |y| < Y_2(x_0) + \delta, y \ne Y_2(x_0)\}$ and has at the point $Y_2(x_0)$, a pole of the second order with the residue equal to

$$\frac{C}{(Y_2(x_0))^{k_2}} - \frac{k_2 C_2}{(Y_2(x_0))^{k_2+1}}$$

with $C_2$ given by (9.14) and some constant $C \in \mathbb{C}$ does not depending on $k_2$. By the residue theorem, it follows that for any $Y_2(x_0) - \varepsilon_2 < \hat{y} < Y_2(x_0)$ and $Y_2(x_0) < y_\delta < Y_2(x_0) + \delta$)

$$\int_{\|y\|=\hat{y}} \frac{F(x_0, y)}{y^{k_2}(1-P(x_0, y))} dy = \frac{2\pi i C_1}{(Y_2(x_0))^{k_2}} + \int_{\|y\|=y_\delta} \frac{F(x_0, y)}{y^{k_2}(1-P(x_0, y))} dy,$$
$$ \int_{\|y\|=\hat{y}} \frac{\partial_x F(x_0, y)}{y^{k_2}(1-P(x_0, y))} dy = \frac{2\pi i \tilde{C}_1}{(Y_2(x_0))^{k_2}} + \int_{\|y\|=y_\delta} \frac{\partial_x F(x_0, y)}{y^{k_2}(1-P(x_0, y))} dy $$

and

$$ \int_{\|y\|=\hat{y}} \frac{F(x_0, y) \partial_x P(x_0, y)}{y^{k_2}(1-P(x_0, y))^2} dy = -2\pi i \left( \frac{C}{(Y_2(x_0))^{k_2}} - \frac{k_2 C_2}{(Y_2(x_0))^{k_2+1}} \right) + \int_{\|y\|=y_\delta} \frac{F(x_0, y) \partial_x P(x_0, y)}{y^{k_2}(1-P(x_0, y))^2} dy $$

Using these relations at the right hand side of (9.15) and (9.16) we obtain

$$ J_1(k) = \frac{C_1}{x_0^{k_1} (Y_2(x_0))^{k_2}} + \frac{1}{(2\pi i)} \int_{\|y\|=y_\delta} \frac{F(x_0, y)}{x_0^{k_1} y^{k_2} (1-P(x_0, y))} dy + \hat{J}_1(k) \quad (9.17) $$

and

$$ \begin{aligned} J_2(k) = & \frac{k_1 C_1}{x_0^{k_1+1} (Y_2(x_0))^{k_2}} - \frac{k_2 C_2}{x_0^{k_1} (Y_2(x_0))^{k_2+1}} + \frac{\tilde{C}}{x_0^{k_1} (Y_2(x_0))^{k_2}} \\ & - \frac{1}{2\pi i x_0^{k_1}} \int_{\|y\|=y_\delta} \frac{\partial_x F(x_0, y)}{y^{k_2}(1-P(x_0, y))} dy + \frac{k_1}{(2\pi i) x_0^{k_1+1}} \int_{\|y\|=y_\delta} \frac{F(x_0, y)}{y^{k_2}(1-P(x_0, y))} dy \\ & - \frac{1}{2\pi i x_0^{k_1}} \int_{\|y\|=y_\delta} \frac{F(x_0, y) \partial_x P(x_0, y)}{y^{k_2}(1-P(x_0, y))^2} dy + \hat{J}_2(k) \end{aligned} \quad (9.18) $$

with some constant $\tilde{C}$ does not depending on $k$. Remark now that by the Cauchy inequality, for $k = (k_1, k_2) \in \mathbb{Z}_+^2$, as $\min\{k_1, k_2\} \to \infty$ and $k/\|k\| \to w = (u, v)$,

$$ \limsup_k \|k\|^{-1} \log \left| \frac{1}{(2\pi i)} \int_{\|y\|=y_\delta} \frac{F(x_0, y)}{x_0^{k_1} y^{k_2} (1-P(x_0, y))} dy \right| \le -(u \log(x_0) + v \log(y_\delta)) \\ < -(u \ln(x_0) + v \ln(Y_2(x_0))), $$

and with the same argument as in the proof of Lemma 9.2,

$$ \limsup_k \|k\|^{-1} \log |\hat{J}_1(k)| \le - \sup_{x_0 \le \hat{x} \le x_0 + \epsilon_1} (u \ln \hat{x} + v \ln(Y_2(\hat{x}))) < -(u \ln(x_0) + v \ln(Y_2(x_0))) $$

where the last relation holds because under our assumptions, $x_P^* < x_0 < x_D(w)$ and by Lemma 9.3, the function $x \mapsto u \ln x + v \ln(Y_2(x))$ is strictly increasing on $[x_P^*, x_D(w)]$. Since for $k = (k_1, k_2) \in \mathbb{Z}_+^2$, as $\|k\| \to \infty$ and $k/\|k\| \to w$,

$$ \lim \|k\|^{-1} \log \left| \frac{F(x_0, Y_2(x_0))}{x_0^{k_1} (Y_2(x_0))^{k_2} \partial_y P(x_0, Y_2(x_0))} \right| = -(u \log(x_0) + v \log(Y_2(x_0))) $$

this proves that as $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w$, the second and the third terms in the right hand side of (9.17) are negligible with respect to the first one. Hence (9.11) is verified, and with the same arguments, from (9.18) one gets (9.13). $\square$

As a consequence of Lemma 9.4 and Theorem 2.3, we obtain the following result.

**Lemma 9.5.** Under the assumptions (A1)-(A3), for any $j \in \mathbb{Z}_+^2 \setminus E_0$ and $w = (u, v) \in \mathbb{S}_+^1$ with $u > u_D(x_d, Y_2(x_d))$, the following assertions hold

i) If either one of the assertions (B0), (B1), (B3), (B4) holds or (B2) holds with $x_d < x_P^{**}$, then for any $k = (k_1, k_2) \in \mathbb{Z}_+^2$, as $\min\{k_1, k_2\} \to \infty$ and $k/\|k\| \to w$,

$$ I_1(j, k) \sim c_1 a_1 \varkappa_1(j) x_d^{-k_1-1} (Y_2(x_d))^{-k_2-1} \quad (9.19) $$
where

$$c_1 = \frac{\phi_1(x, y) - 1}{\partial_y P(x, y)} \Big|_{(x,y)=(x_d, Y_2(x_d))} > 0 \qquad (9.20)$$

and $a_1 > 0$ is given by (2.41).

ii) If (B5) holds with $x_d < x_P^{**}$, then for $k = (k_1, k_2) \in \mathbb{Z}_+^2$, as $\min\{k_1, k_2\} \to \infty$ and $k/\|k\| \to w$,

$$I_1(j, k) \sim \frac{c_1 a_3 \varkappa_2(j) k_1}{x_d^{k_1+2} (Y_2(x_d))^{k_2+1}} + \frac{\tilde{c}_1 a_3 \varkappa_2(j) k_2}{x_d^{k_1+1} (Y_2(x_d))^{k_2+2}} \qquad (9.21)$$

where $c_1 > 0$ is given by (9.20),

$$\tilde{c}_1 = \frac{\partial_x P(x_d, Y_2(x_d))}{(\partial_y P(x_d, Y_2(x_d)))^2} \Big|_{(x,y)=(x_d, Y_2(x_d))}$$

and $a_3 > 0$ is given by (2.47).

iii) If (B6) holds, then for $k = (k_1, k_2) \in \mathbb{Z}_+^2$, as $\min\{k_1, k_2\} \to \infty$ and $k/\|k\| \to w$,

$$I_1(j, k) \sim c_1 a_5 \varkappa_2(j) x_d^{-k_1-1} (Y_2(x_d))^{-k_2-1} \qquad (9.22)$$

where $c_1 > 0$ is given by (9.20) and $a_5 > 0$ is given by (2.53).

**Proof.** Indeed, when either, one of the conditions (B0), (B1), (B3) or (B4) is satisfied, or (B2) holds with $x_d < x_P^{**}$, by the first assertion of Theorem 2.3 and using Proposition 7.13, one gets that for some $\varepsilon > 0$, the function $x \mapsto H_j(x, 0)$ can be extended as an analytic function to the set $B(0, x_d + \varepsilon) \setminus \{x_d\}$ and has a simple pole at the point $x_d$ with the residue

$$-a_1 \varkappa_1(j) = -\varkappa_1(j) \left( \frac{d}{dx} \phi_1(x_d, Y_1(x_d)) \right)^{-1} < 0.$$

Since the function $(x, y) \mapsto \phi_1(x, y) - P(x, y)$ is analytic in a neighborhood of the set $\Omega(D)$, this implies that the function $(x, y) \mapsto F(x, y) = (x_d - x)(\phi_1(x, y) - P(x, y))H_j(x, 0)$ can be extended as an analytic function to a neighborhood of the set $\{(x, y) \in \Omega(D) : |x| < x_d + \varepsilon\}$ by letting

$$F(x_d, y) = (\phi(x_d, y) - P(x_d, y)) a_1 \varkappa_1(j).$$

When either, one of the conditions (B0), (B1), (B3) or (B4) is satisfied, or (B2) holds with $x_d < x_P^{**}$, one has always $x_P^* < x_d = x^{**} < x_P^{**}$, and consequently, by Lemma 4.1, $Y_2(x_d) > Y_1(x_d)$ and by Lemma 4.1, $\phi_1(x_d, Y_1(x_d)) = 1$. Since the function $y \mapsto \phi_1(x_d, y)$ is strictly increasing, it follows that $\phi_1(x_d, Y_2(x_d)) > 1$. Since by Theorem 2.3, $a_1 > 0$, we obtain therefore

$$F(x_d, Y_2(x_d)) = (\phi_1(x_d, Y_2(x)) - P(x_d, Y_2(x_d))) a_1 \varkappa_1(j) = (\phi_1(x_d, Y_2(x)) - 1) a_1 \varkappa_1(j) > 0.$$

By Lemma 9.4, it follows that for any $w = (u, v) \in \mathbb{S}_+^1$ such that $x_D(u) > x_d$, and as $\min\{k_1, k_2\} \to +\infty$, (9.19) holds with $c_1 > 0$ given by (9.20). Since for $w = (u, v) \in \mathbb{S}_+^2$ the inequality $x_D(u) > x_d$ is equivalent to the inequality $u > u_D(x_d, Y_2(x_d))$, the first assertion of our lemma is therefore proved.

Suppose now that (B5) holds with $x_d < x_P^{**}$. Then, by Theorem 2.3 and using Proposition 7.13, one gets that for some $\varepsilon > 0$, the function $x \mapsto H_j(x, 0)$ can be extended as an analytic function to the set $B(0, x_d + \varepsilon) \setminus \{x_d\}$ and has a pole of the second order at the point $x_d$ with

$$\lim_{x \to x_0} (x_d - x)^2 H_j(x, 0) = a_3 \varkappa_2(j) > 0$$
where $a_3 > 0$ is given by (2.47). With the same arguments as above, it follows that the function $(x, y) \mapsto F(x, y) = (x_d - x)^2(\phi_1(x, y) - 1)H_j(x, 0)$ can be extended as an analytic function to a neighborhood of the set $\{(x, y) \in \Omega(D) : |x| < x_d + \varepsilon\}$ by letting $F(x_d, y) = (\phi_1(x_d, y) - 1)a_3\varkappa_2(j)$ and that the extended function $F$ satisfies the assumptions of Lemma 9.4 with $x_0 = x_d$ and $F(x_0, Y_2(x_0)) = a_3\varkappa_2(j)(\phi_1(x_d, Y_2(x_d)) - 1) > 0$. Hence, using (9.13) we obtain (9.21) with $c_1 > 0$ given by (9.20). The second assertion of our lemma is therefore also proved.

The proof of the third assertion of our lemma is exactly the same as the proof the first assertion, with an only difference that now, one should use the fifth assertion of Theorem 2.3 instead of the first one. □

Remark finally that if we exchange the roles of $x$ and $y$, then with the same arguments as above we obtain the following result.

**Lemma 9.6.** Under the assumptions (A1)-(A3), for any $j \in \mathbb{Z}_+^2 \setminus E_0$ and $w = (u, v) \in \mathbb{S}_+^1$ with $v > v_D(X_2(y_d), y_d)$, the following assertions hold

i) If either one of the assertions (B0), (B1), (B5), (B6) holds or (B2) holds with $y_d < y_P^{**}$, then for any $k = (k_1, k_2) \in \mathbb{Z}_+^2$, as $\min\{k_1, k_2\} \to \infty$ and $k/\|k\| \to w$,

$$I_2(j, k) \sim c_2 b_2 \varkappa_2(j) (X_2(y_d))^{-k_1-1} (y_d)^{-k_2-1} \quad (9.23)$$

where

$$c_2 = (\phi_2(x, y) - 1) (\partial_x P(x, y))^{-1} \Big|_{(x,y)=(X_2(y_d), y_d)} > 0 \quad (9.24)$$

and

$$a_2 = \left( \frac{d}{dy} \phi_2(X_1(y), y) \right)^{-1} \Big|_{y=y_d} \quad (9.25)$$

ii) If (B3) holds with $y_d < y_P^{**}$, then for $k = (k_1, k_2) \in \mathbb{Z}_+^2$, as $\min\{k_1, k_2\} \to \infty$ and $k/\|k\| \to w$,

$$I_2(j, k) \sim \frac{c_2 b_3 \varkappa_1(j) k_2}{(X_2(y_d))^{k_1+1} y_d^{k_2+2}} + \frac{\tilde{c}_2 b_3 \varkappa_1(j) k_1}{(X_2(y_d))^{k_1+2} y_d^{k_2+1}} \quad (9.26)$$

with $c_2 > 0$ given by (9.24),

$$a'_3 = (\phi_1(x, y) - 1) \left( \left( \frac{d}{dy} \phi_2(X_1(y), y) \right) \frac{d}{dx} \phi_1(x, Y_1(x)) \frac{d}{dy} X_1(y) \right)^{-1} \Big|_{(x,y)=(X_2(y_d), y_d)} > 0 \quad (9.27)$$

and some $\tilde{c}_2 \in \mathbb{R}$ do not depending on $j \in \mathbb{Z}_+^2 \setminus E_0$.

iii) If (B4) holds, then for $k = (k_1, k_2) \in \mathbb{Z}_+^2$, as $\min\{k_1, k_2\} \to \infty$ and $k/\|k\| \to w$,

$$I_2(j, k) \sim c_2 a'_5 \varkappa_1(j) (X_2(y_d))^{-k_1-1} (y_d)^{-k_2-1} \quad (9.28)$$

with $c_2$ given by (9.24) and

$$a'_5 = (\phi_1(x, y) - 1) \left( (1 - \phi_2(x, y)) \frac{d}{dx} \phi_1(x, Y_1(x)) \frac{d}{dy} X_1(y) \right)^{-1} \Big|_{(x,y)=(X_2(y_d), y_d)} > 0 \quad (9.29)$$
## 9.5 Proof of the assertions i) and ii) of Theorem 2.6

For $I_1(j, k)$ the exact asymptotics as $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w = (u, v) \in \mathbb{S}_+^1$ were obtained only in the case where $u > u_D(x_d, Y_2(x_d))$. In the following lemma, we show that this inequality always holds when $w \in W_1$. The definitions 3-6 give that the set $W_1$ is empty if either, one of the cases (B5), or (B6) occurs or, (B2) and $x_d=x_P^{**}$ hold, we can assume that either one of cases (B0), (B1), (B3) or (B4) occurs or (B2) and $x_d < x_P^{**}$ hold.

**Lemma 9.7.** If conditions (A1)-(A3) hold and if either, one of cases (B0), (B1), (B3) or (B4) holds, or (B2) and $x_d<x_P^{**}$ hold, then for any $w \in W_1$,

$$u > u_D(x_d, Y_2(x_d)). \qquad (9.30)$$

*Proof.* Suppose first that (B0) holds. Then by Proposition 2.2 and the definition of the points $x_d$ and $y_d$ (see (2.33) and (2.34)),

$$X_1(y_d) < x_d = x^{**} < X_2(y_d), \quad Y_1(x_d) < y_d = y^{**} < Y_2(x_d). \qquad (9.31)$$

In this case $W_1 = \{w = (u, v) \in \mathbb{S}_+^1 : u > u_c\}$ where $w_c = (u_c, v_c)$ is the only point in $\mathbb{S}_+^1$ such that

$$u_c \ln(x_d) + v_c \ln(Y_2(x_d)) = u_c \ln(X_2(y_d)) + v_c \ln(y_d), \qquad (9.32)$$

or equivalently, such that

$$(\ln(X_2(y_d)) - \ln(x_d))u_c = (\ln(Y_2(x_d)) - \ln(y_d))v_c. \qquad (9.33)$$

Since by (9.31), $X_2(y_d) > x_d$ and $Y_2(x_d) > y_d$, the last relation implies that $u_c > 0$ and $v_c > 0$, and consequently, if $u_D(x_d, Y_2(x_d)) \le 0$, one gets

$$u_D(x_d, Y_2(x_d)) \le u_c < u, \quad \forall w = (u, v) \in W_1.$$

When (B0) holds and $u_D(x_d, Y_2(x_d)) \le 0$, our lemma is therefore proved.

Consider now the case when (B0) holds and $u_D(x_d, Y_2(x_d)) > 0$. For $\hat{w} = (\hat{u}, \hat{v}) = (u_D(x_d, Y_2(x_d)), v_D(x_d, Y_2(x_d)))$, by the definition of the mapping $w \mapsto (x_D(w), y_d(w))$, the point $(x_d, Y_2(x_d))$ is the only point in the set $D$ where the function $(x, y) \mapsto \hat{u} \ln(x) + \hat{v} \ln(y)$ achieves its maximum over the set $D$. Since in the case (B0), $(x_d, Y_2(x_d)) \ne (X_2(y_d), y_d)$, it follows that for $\hat{w} = (\hat{u}, \hat{v}) = (u_D(x_d, Y_2(x_d)), v_D(x_d, Y_2(x_d)))$,

$$\hat{u} \ln(x_d) + \hat{v} \ln(Y_2(x_d)) > \hat{u} \ln(X_2(y_d)) + \hat{v} \ln(y_d)$$

or equivalently that

$$(\ln(X_2(y_d)) - \ln(x_d))\hat{u} < (\ln(Y_2(x_d)) - \ln(y_d))\hat{v}.$$

Since $X_2(y_d) > x_d$ and $Y_2(x_d) > y_d$, and since we assumed that $\hat{u} = u_D(x_d, Y_2(x_d)) > 0$, the last inequality shows that $\hat{v} = v_D(x_d, Y_2(x_d)) > 0$ and consequently, $\hat{v} = \sqrt{1 - \hat{u}^2}$ and

$$(\ln(X_2(y_d)) - \ln(x_d))\hat{u} < (\ln(Y_2(x_d)) - \ln(y_d))\sqrt{1 - \hat{u}^2}.$$

Finally, we have already proved that $u_c > 0$ and $v_c > 0$, and in this case, relation (9.33) is equivalent to

$$(\ln(X_2(y_d)) - \ln(x_d))u_c = (\ln(Y_2(x_d)) - \ln(y_d))\sqrt{1 - u_c^2}.$$

Since the function $u \mapsto (\ln(X_2(y_d)) - \ln(x_d))u$ is strictly increasing and the function $u \mapsto (\ln(Y_2(x_d)) - \ln(y_d))\sqrt{1 - u_c^2}$ is decreasing on the segment $[0, 1]$, the last two relations show that $u_c > \hat{u} = u_D(x_d, Y_2(x_d))$, and consequently using the definition of $W_1$ one gets $u > u_c > u_D(x_d, Y_2(x_d))$ for all $u \in W_1$. If (B0) holds, Lemma 9.7 is therefore proved.
When either, (B1) holds, or (B2) and $x_d<x_P^{**}$ hold, relation (9.30) follows directly from the definition of the set $W_1$ (see Definition 4 and Definition 5).

Consider now the case when (B3) holds. With definition 6, $W_1 = \{w = (u, v) \in \mathbb{S}_+^1: u > 0\}$ and by Proposition 2.2 and the definition of the points $x_d$ and $y_d$ (see (2.33) and (2.34)), one has $(x_d, Y_2(x_d)) \in S_{12}$. By Lemma 4.1 and the definition of the mapping $(x, y) \mapsto w_D(x, y)$, it follows that

$$u_D(x_d, Y_2(x_d)) = \frac{x_d \partial_x P(x_d, y_d)}{\sqrt{(x_d \partial_x P(x_d, y_d))^2 + (y_d \partial_y P(x_d, y_d))^2}} \le 0.$$

and consequently, for any $w = (u, v) \in W_1$, (9.30) holds.

Suppose now that (B4) holds. Then, with the Definition 6, $W_1 = \mathbb{S}_+^1$ and by Proposition 2.2 and the definition of the points $x_d$ and $y_d$ (see (2.33) and (2.34)) and using (2.9) one gets

$$X_1(y_d) = x_d = x^{**} < x_P^{**}, \quad y_P^{*} < y^{*} \le 1 < y_d = Y_2(x_d) < y^{**} \le y_P^{**}, \quad (x_d, y_d) \in S_{12}. \quad (9.34)$$

By Lemma 4.1 and the definition of the mapping $(x, y) \mapsto w_D(x, y)$, it follows that for any $w = (u, v) \in W_1$,

$$u_D(x_d, Y_2(x_d)) = u_D(x_d, y_d) = u_D(X_1(x_d), y_d) = \frac{x_d \partial_x P(x_d, y_d)}{\sqrt{(x_d \partial_x P(x_d, y_d))^2 + (y_d \partial_y P(x_d, y_d))^2}} < 0 \le u,$$

and consequently, (9.30) holds. □

As a straightforward consequence of Lemma 9.5 and Lemma 9.7, one gets

**Corollary 9.2.** Under the assumptions of Theorem 2.6, for any $j \in \mathbb{Z}_+^2 \setminus E_0$ and $w \in W$, as $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w$, (9.19) holds.

Now we will show that for any $w = (u, v) \in W$, as $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w$, the terms $I_0(j, k)$ and $I_2(j, k)$ are negligible with respect to $I_1(j, k)$. For this we will use the large deviation estimates for $I_0(k, j)$, and

- the large deviation estimates of $I_2(j, k)$ when

$$\limsup_k \frac{1}{\|k\|} \ln I_2(j, k) < \lim \frac{1}{\|k\|} \ln I_1(j, k),$$

- the exact asymptotic of $I_2(j, k)$ when

$$\limsup_k \frac{1}{\|k\|} \ln I_2(j, k) = \lim \frac{1}{\|k\|} \ln I_1(j, k).$$

To compare the limit $\lim_k \ln I_1(j, k)/\|k\|$ with the large deviation estimates for $\limsup_k \ln I_0(j, k)/\|k\|$ and $\limsup_k \ln I_2(j, k)/\|k\|$, the following lemma will be useful.

**Lemma 9.8.** If conditions (A1)-(A3) hold and let $w=(u, v) \in W_1$, if one of the assertions holds,

- one of the conditions (B0), (B1) or (B3) holds,

- (B2) and $x^{**}<x_P^{**}$ hold,

- (B4) holds and $u > 0$,

then

$$\max_{(x,y) \in D, y \le y_d} \left( u \ln(x) + v \ln(y) \right) > u \ln(x_d) + v \ln(Y_2(x_d)). \quad (9.35)$$
Proof. Consider first the case when (B0) holds. We have, see the proof of Lemma 9.8, $X_2(y_d) > x_d$ and $Y_2(x_d) > y_d$, the point $(w_c = (u_c, v_c) \in \mathbb{S}_+^2$ satisfies (9.33) and $W_1 = \{w = (u, v) \in \mathbb{S}_+^1 : u > u_c\}$. For $w = (u, v) \in \mathbb{S}_+^1$ with $u > u_c$ (and consequently also with $v < v_c$), it follows that

$$ (\ln(X_2(y_d)) - \ln(x_d))u > (\ln(Y_2(x_d)) - \ln(y_d))v $$

or equivalently,

$$ u \ln(X_2(y_d)) + v \ln(y_d) > u \ln(x_d) + v \ln(Y_2(x_d)). \quad (9.36) $$

Since the point $(X_2(y_d), y_d)$ belongs to the set $\{(x, y) \in D : y \le y_d\}$, this proves that for any $w = (u, v) \in \mathbb{S}_+^1$ with $u > u_c$,

$$ \max_{(x,y) \in D, y \le y_d} \left( u \ln(x) + v \ln(y) \right) \ge u \ln(X_2(y_d)) + v \ln(y_d) > u \ln(x_d) + v \ln(Y_2(x_d)), \quad (9.37) $$

and consequently, (9.35) holds.

Consider now the case when (B1) holds, we have the relation

$$ W_1 = \{w = (u, v) \in \mathbb{S}_+^1 : u > u_D(x_d, Y_2(x_d))\}, $$

and, by Proposition 2.2 and the definition of the points $x_d$ and $y_d$ (see (2.33) and (2.34)),

$$ X_1(y_d) < x_d = x^{**} = X_2(y_d), \quad Y_1(x_d) < y_d = y^{**} = Y_2(x_d) \quad (9.38) $$

and $(x_d, Y_2(x_d)) = (X_2(y_d), y_d) \in S_{22}$. Hence, for any $w = (u, v) \in W_1$,

$$ x_D(w) > x_d = X_2(y_d) \quad \text{and} \quad y_D(w) < y_d = Y_2(x_d). \quad (9.39) $$

The last relations show that the point $(x_D(w), y_D(w))$ belongs to the set $\{(x, y) \in D : y \le y_d\}$ and is not equal to $(x_d, Y_2(x_d))$. Since the point $(x_D(w), y_D(w))$ is an only point in $D$ where the function $(x, y) \mapsto u \ln(x) + x \ln(y)$ achieves its maximum over $D$, it follows that

$$ \max_{(\hat{x}, \hat{y}) \in D, \hat{y} \le y_d} \left( u \ln(\hat{x}) + v \ln(\hat{y}) \right) = u \ln(x_D(w)) + v \ln(y_D(w)) > u \ln(x_d) + v \ln(Y_2(x_d)), \quad (9.40) $$

and consequently, when the assertion (B1) holds, relation (9.35) is also proved.

Consider now the case when (B2) and $x_d < x_P^{**}$ hold. By Proposition 2.2 and the definition of the points $x_d$ and $y_d$ (see (2.33) and (2.34)),

$$ x_d = x^{**} > X_2(y_d), \quad y_d = y^{**} > Y_2(x_d), \quad (x_d, Y_2(x_d)), (X_2(y_d), y_d) \in S_{22}, \quad (9.41) $$

and with the definition of $W_1$,

$$ W_1 = \{w = (u, v) \in \mathbb{S}_+^1 : u > u_D(x_d, Y_2(x_d))\}. $$

Hence, in this case, for any $w = (u, v) \in W_1$,

$$ x_D(w) > x_d > X_2(y_d) \quad \text{and} \quad y_D(w) < Y_2(x_d) < y_d. $$

These relations show that the point $(x_D(w), y_D(w))$ belongs to the set $\{(x, y) \in D : y \le y_d\}$ and is not equal to $(x_d, Y_2(x_d))$, and consequently, using exactly the same arguments as in the previous case, one gets (9.35).

Suppose now that (B3) and $y_d < y_P^{**}$ hold. By Proposition 2.2 and the definition of the points $x_d$ and $y_d$ (see (2.33) and (2.34)), we have

$$ X_1(y_d) = x_d = x^{**} < x_P^{**}, \quad y_d = y^{**} = Y_2(x_d), \quad (x_d, y_d) = (x_d, Y_2(x_d)) \in S_{12} \quad (9.42) $$
and

$$W_1 = \{w = (u, v) \in \mathbb{S}_+^1 : u > 0\}.$$

Since we assume that $y_d < y_P^{**}$ and by (2.9), $y^{**} > y^* \ge y_P^*$, by Lemma 4.1 and the definition of the mapping $(x, y) \mapsto w_D(x, y) = (u_D(x, y), v_D(x, y))$, from (9.42) it follows that

$$x_d = X_1(y_d) < X_2(y_d) \quad \text{and} \quad u_D(x_d, Y_2(x_d)) = u_D(x_d, y_d) < 0. \qquad (9.43)$$

Hence, in this case, for any $w = (u, v) \in W_1$,

$$\begin{aligned} u \ln(x_d) + x \ln(Y_2(x_d)) &= u \ln(x_d) + v \ln(y_d) < u \ln(X_2(y_d)) + v \ln(y_d) \\ &\le \max_{(x,y) \in D, y \le y_d} (u \ln(x) + v \ln(y)) \quad (9.44) \end{aligned}$$

and consequently, (9.35) holds.

Suppose now that (B3) holds and $y_d = y_P^{**}$. Then $W_1 = \{w = (u, v) \in \mathbb{S}_+^1 : u > 0\}$ and (9.42) holds, but now, instead of (9.43) one has

$$x_d = X_1(y_d) = X_2(y_d), \quad y_d = y^{**} = y_P^{**} \quad \text{and} \quad w_D(x_d, y_d) = (0, 1).$$

Hence, in this case, for any $w = (u, v) \in W_1$,

$$x_D(w) > x_d \quad \text{and} \quad y_D(w) < y_d. \qquad (9.45)$$

These relations show that the point $(x_D(w), y_D(w))$ belongs to the set $\{(x, y) \in D : y \le y_d\}$ and is not equal to $(x_d, Y_2(x_d))$ and consequently, using exactly the same arguments as in the case (B2), one gets (9.35).

Consider finally the case when (B4) holds. In this case, $W_1 = \mathbb{S}_+^1$ and by Proposition 2.2 and the definition of the points $x_d$ and $y_d$ (see (2.33) and (2.34))

$$X_1(y_d) = x_d = y^{**} < x_P^{**}, \quad y^* \le 1 < y_d = Y_2(x_d) < y^{**} \le y_P^{**}, \quad (x_d, y_d) \in S_{12}. \qquad (9.46)$$

Hence, in this case, (9.43) holds and consequently, using exactly the same arguments as above, one gets (9.35) for any $w = (u, v) \in W_1$ with $u > 0$. $\square$

Now we are ready to complete the proof of the first assertion of Theorem 2.6: Since the right hand side of (2.75) does not depend on $w \in W_1$, it is sufficient to show that for any $j \in \mathbb{Z}_+^2 \setminus E_0$ and $w = (u, v) \in W_1$, (2.75) holds as $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w$.

Consider first the case when one of the following assertion holds

- one of the conditions (B0), (B1) or (B3) holds

- (B2) and $x^{**} < x_P^{**}$ hold,

- (B4) holds and $u > 0$.

By using Lemma 9.2 and Lemma 9.8, as $\|k\| \to +\infty$ and $k/\|k\| \to w$, one gets

$$\begin{aligned} \limsup_k \|k\|^{-1} \log |I_0(j, k) + I_2(j, k)| &\le - \max_{(\hat{x}, \hat{y}) \in D, \hat{y} \le y_d} (u \ln(\hat{x}) + v \ln(\hat{y})) \\ &< -u \ln(x_d) + v \ln(Y_2(x_d)). \end{aligned} \qquad (9.47)$$

By Corollary 9.2, for any $w = (u, v) \in W_1$ and $j \in \mathbb{Z}_+^2 \setminus E_0$, as $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w$,

$$I_1(j, k) \sim c_1 a_1 \chi_1(j) x_d^{-k_1-1} (Y_2(x_d))^{-k_2-1} \qquad (9.48)$$
where $c_1 > 0$ is given by (9.20) and $a_1 > 0$ is given by (2.41).

Comparison of (9.48) with (9.47) shows that the terms $I_0(j, k)$ and $I_2(j, k)$ in (9.2) are negligible with respect to $I_1(j, k)$, and consequently, from the integral representation (9.2) and using (9.48) one gets (2.75).

The set of directions $W_1$ is empty in each of the following cases:

- (B2) and $x_d=x_P^{**}$ hold;

- (B5) or (B6) holds.

Hence, to complete the proof of the first assertion of Theorem 2.6 it is sufficient now to prove (2.75) when (B4) holds and $w=(0, 1)$. By Proposition 2.2 and the definition of the points $x_d$ and $y_d$ (see (2.33) and (2.34)) one has

$$X_1(y_d) = x_d = x^{**} < x_P^{**}, \quad y^* \le 1 < y_d = Y_2(x_d) < y^{**} \le y_P^{**}, \quad (x_d, y_d) \in S_{12}. \quad (9.49)$$

and the definition of the point $y_P^{**}$ and the mapping $w \mapsto (x_D(w), y_D(w))$ give, for $w = (0, 1)$,

$$(x_D(0, 1), y_D(0, 1)) = (X_1(y_P^{**}), y_P^{**}). \quad (9.50)$$

Hence, in this case, $y_D(0, 1) = y_P^{**} > y_d$, and consequently, also $1 > v_D(X_2(y_d), y_d)$. By Lemma 9.6 applied with $w = (0, 1)$, it follows that there is $C > 0$ such that for any $j \in \mathbb{Z}_+^2 \setminus E_0$ and $k = (k_1, k_2) \in \mathbb{Z}_+^2$, and as $\min\{k_1, k_2\} \to +\infty$, $k/\|k\| \to (0, 1)$

$$I_2(j, k) \sim C \chi_1(j)(X_2(y_d))^{-k_1-1}(y_d)^{-k_2-1}. \quad (9.51)$$

With Lemma 9.2 applied with $w = (0, 1)$, the definition of the point $y_P^{**}$, and using relation (9.49),

$$\limsup_k \|k\|^{-1} \ln |I_0(j, k)| \le - \max_{(x,y) \in D} \ln(y) = - \ln(y_P^{**}) < - \ln(y_d). \quad (9.52)$$

Remark that by relation (9.49) and Corollary 9.2 applied with $w = (0, 1) \in W_1$, for any $j \in \mathbb{Z}_+^2 \setminus E_0$, as $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to (0, 1)$,

$$I_1(j, k) \sim c_1 a_1 \chi_1(j) x_d^{-k_1-1} (Y_2(x_d))^{-k_2-1} = c_1 a_1 \chi_1(j) x_d^{-k_1-1} y_d^{-k_2-1} \quad (9.53)$$

where $c_1 > 0$ is given by (9.20) and $a_1 > 0$ is given by (2.41).

A comparison of relation (9.52) with relations (9.51) and (9.53) shows that for any $j \in \mathbb{Z}_+^2 \setminus E_0$ and $k = (k_1, k_2) \in \mathbb{Z}_+^2$, and as $\min\{k_1, k_2\} \to +\infty$, $k/\|k\| \to (0, 1)$, the term $I_0(j, k)$ is negligible with respect to $I_1(j, k) + I_2(j, k)$ in (9.2), and consequently, using (9.51) and (9.53) one gets

$$g(j, k) \sim I_1(j, k) + I_2(j, k) \sim c_1 a_1 \chi_1(j) x_d^{-k_1-1} y_d^{-k_2-1} + C \chi_1(j) (X_2(y_d))^{-k_1-1} (y_d)^{-k_2-1}.$$

Since by (9.49), $x_d = X_1(y_d) < X_2(y_d)$, this proves (2.75) with $b_1 = c_1 a_1$.

The first assertion of Theorem 2.6 is therefore proved. The proof of the second assertion is the same by exchanging the roles of $x$ and $y$.

## 9.6 Proof of the assertion iii) of Theorem 2.6

Suppose first that (B0) holds. In this case, see the proof of Lemma 9.7,

$$u_c > u_D(x_d, Y_2(x_d)) \quad \text{and} \quad v_c > v_D(X_2(y_d), y_d). \quad (9.54)$$

Consider $j \in \mathbb{Z}_+^2 \setminus E_0$ and let $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w_c$. Then using the first relation of (9.54), by Lemma 9.5 applied for $w = w_c = (u_c, v_c)$, we get

$$I_1(j, k) \sim c_1 a_1 \chi_1(j) x_d^{-k_1-1} (Y_2(x_d))^{-k_2-1}, \quad (9.55)$$
where $c_1 > 0$ and $a_1 > 0$ given respectively by (9.20) and (2.41), and using the second relation of (9.54), by Lemma 9.6 applied for $w = w_c = (u_c, v_c)$, we obtain

$$I_2(j, k) \sim c_2 a_2 \varkappa_2(j) (X_2(y_d))^{-k_1-1} y_d^{-k_2-1} \quad (9.56)$$

where $c_2 > 0$ and $a_2 > 0$ are defined respectively by (9.24) and (9.25). Comparison of (9.55) and (9.56) with (9.7) shows that the term $I_0(j, k)$ in (9.2) is negligible with respect to $I_1(j, k) + I_1(j, k)$, and consequently, using (9.2) together with (9.55) and (9.56) one gets (2.79) with $b_1 = c_1 a_1$ and $b_2 = c_2 a_2$. .

## 9.7 Proof of the assertions iv) and v) of Theorem 2.6

Suppose that (B3) holds, $y_d < y_P^{**}$ and let $w = (0, 1)$. By Proposition 2.2 and the definition of the points $x_d$ and $y_d$, one has $x_P^* < x_d = x^{**} < x_P^{**}$, $(x_d, y_d) = (x_d, Y_2(x_d)) = (X_1(y_d), y_d) \in S_{12}$ and $y_P^* < y_d = y^{**} < y_P^{**}$. By Lemma 4.1 and the definition of the mapping $(x, y) \mapsto w_D(x, y) = (u_D(x, y), v_D(x, y))$, it follows that

$$x_d = X_1(y_d) < X_2(y_d), \quad (9.57)$$

$$u_D(x_d, Y_2(x_d)) < 0 \quad \text{and} \quad v_D(X_2(y_d), y_d) < 1. \quad (9.58)$$

Consider now $j \in \mathbb{Z}_+^2 \setminus E_0$ and let $k \in \mathbb{Z}_+^2$, $\min\{k_1, k_2\} \to +\infty$ and $k/\|k\| \to w = (1, 0)$. Then by Lemma 9.2 and since $y_d < y_P^{**}$, one gets

$$\limsup_k \|k\|^{-1} \ln |I_0(j, k)| \le - \max_{(x,y) \in D} \ln(y) = - \ln(y_P^{**}) < - \ln(y_d), \quad (9.59)$$

by Lemma 9.5, from the first relation of (9.58) it follows

$$I_1(j, k) \sim c_1 a_1 \varkappa_1(j) x_d^{-k_1-1} (Y_2(x_d))^{-k_2-1} = c_1 a_1 \varkappa_1(j) x_d^{-k_1-1} y_d^{-k_2-1}, \quad (9.60)$$

where $c_1 > 0$ and $a_1 > 0$ are given respectively by (9.20) and (2.41), and by the second assertion of Lemma 9.6, from the second relation of (9.58) it follows that there exists $\tilde{c}_2 \in \mathbb{R}$ do not depending on $j \in \mathbb{Z}_+^2 \setminus E_0$ such that

$$I_2(j, k) \sim \frac{c_2 a'_3 \varkappa_1(j) k_2}{(X_2(y_d))^{k_1+1} y_d^{k_2+2}} + \frac{\tilde{c}_2 a'_3 \varkappa_1(j) k_1}{(X_2(y_d))^{k_1+2} y_d^{k_2+1}} \quad (9.61)$$

where $c_2 > 0$ and $a'_3 > 0$ are given respectively by (9.24) and (9.27).

Comparison of (9.59) with (9.60) shows that the term $I_0(j, k)$ is negligible with respect to $I_1(j, k) + I_2(j, k)$ in (9.2). Using therefore (9.2) together with (9.60) and (9.61) we obtain

$$g(j, k) \sim I_1(j, k) + I_2(j, k) \\ \sim \frac{c_1 a_1 \varkappa_1(j)}{x_d^{k_1+1} y_d^{k_2+1}} + \frac{c_2 a'_3 \varkappa_1(j) k_2}{(X_2(y_d))^{k_1+1} y_d^{k_2+2}} + \frac{\tilde{c}_2 b_3 \varkappa_1(j) k_1}{(X_2(y_d))^{k_1+2} y_d^{k_2+1}} \quad (9.62)$$

Finally, by relation (9.57) we have $x_d < X_2(y_d)$, hence, in relation (9.62), the last term is negligible with respect to the first one and consequently, relation (2.80) holds with $b_1 = c_1 a_1$ and $b_3 = c_2 a'_3$.

The assertion iv) of Theorem 2.6 is therefore also proved. The proof of the assertion v) is exactly the same (it is sufficient to exchange the roles of the first and the second coordinates of the random walk $(Z(n))$).
## 9.8 Proof of the assertion vi) of Theorem 2.6

To prove the last assertion of Theorem 2.6 it is sufficient to show that for any $w \in W_0$, (2.84) holds uniformly with respect to $w_m = m/\|m\|$ in some neighborhood of $w$. For this we use Proposition 1 of [18]. In our setting, this result gives the following lemma.

**Lemma 9.9.** Suppose that the Assumption (A1) is satisfied and let $w = (u, v) \in \mathbb{S}_+^1$ and $\varepsilon>0$ and a function $(x, y) \mapsto F(x, y)$ be analytic in the polycircular set

$$ \{(x, y) \in \mathbb{C}^2 : |x - x_D(w)| < \varepsilon, |y - y_D(w)| < \varepsilon\}, \qquad (9.63) $$

and not vanishing at the point $(x_D(w), y_D(w))$. Then the integrals

$$ I(m) = \frac{1}{(2\pi i)^2} \int_{|x|=\hat{x}} \int_{|y|=\hat{y}} \frac{F(x, y)}{x^{m_1+1} y^{m_2+1} (1-P(x, y))} dxdy, \quad m = (m_1, m_2) \in \mathbb{Z}_+^2, \qquad (9.64) $$

are well defined and does not depend on the point $(\hat{x}, \hat{y})$ on the set

$$ \{(\hat{x}, \hat{y}) \in \mathring{D} : |\hat{x} - x_D(w)| < \varepsilon, |\hat{y} - y_D(w)| < \varepsilon\}, \qquad (9.65) $$

and as $\|m\| \to +\infty$, uniformly with respect to $w_m = m/\|m\|$ in some neighborhood of $w$,

$$ I(m) \sim \frac{F(x_D(w_k), y_D(w_k)) \sqrt{w_k^{\perp} \cdot Q(w_k) w_k^{\perp}}}{(2\pi \|k\|)^{1/2} \|m(w_k)\|^{-1} (x_D(w_k))^{k_1} (y_D(w_k))^{k_2}} \qquad (9.66) $$

By Corollary 9.1, for any $(\hat{x}, \hat{y}) \in \mathring{D}$, with $\hat{x} < x_d$ and $\hat{y} < y_d$, we have

$$ g(j, k) = \frac{1}{(2\pi i)^2} \int_{|x|=\hat{x}} \int_{|y|=\hat{y}} \frac{F_j(x, y)}{x^{k_1+1} y^{k_2+1} (1-P(x, y))} dxdy. \qquad (9.67) $$

with

$$ F_j(x, y) = L_j(x, y) + (\phi_1(x, y) - P(x, y))H_j(x, 0) + (\phi_2(x, y) - P(x, y))H_j(0, y). $$

Remark moreover that the set of directions $W_0$ is non-empty if and only if (B2) holds, and that in this case, for any $w = (u, v) \in W_0$, one has

$$ x_D(w) < x_d, \quad y_D(w) < y_d. \qquad (9.68) $$

By Theorem 2.1, it follows that for some neighborhood $V(x_D(w), y_D(w))$ of the point $(x_D(w), y_D(w))$ in $\mathbb{R}^2$, the function $(x, y) \mapsto F_j(x, y) = L_j(x, y) + (\phi_1(x, y) - 1)H_j(x, 0) + (\phi_2(x, y) - 1)H_j(0, y)$ is analytic in the polycircular set $\{(x, y) \in \mathbb{C}^2 : (|x|, |y|) \in V(x_D(w), y_D(w))\}$, and by Proposition 7.13,

$$ F_j(x_D(w), y_D(w)) = \varkappa_{(x_D(w), y_D(w))}(j) > 0, \quad \forall j \in \mathbb{Z}_+^2 \setminus E_0. $$

For any $w \in W_0$ and $j \in \mathbb{Z}_+^2 \setminus E_0$, the conditions of Lemma 9.9 are therefore satisfied with $F=F_j$, and consequently relation (2.84) holds uniformly with respect to $w_k=k/\|k\|$ in some neighborhood of $w$.

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A Relations with Positive Recurrence and Transience Conditions

This section discusses the conditions (B0)-(B7) defining our classification with transience and positive recurrence conditions of these random walks. As we will see, the regions of our classification are in fact not defined in terms of positive recurrence or transience. Transience and positive recurrence are possible in several regions.

Throughout this section, $\mu_0$, $\mu_1$ or $\mu_2$ are assumed to be stochastic, i.e. probability distributions. Before getting these results, note that Assumption (A1)’(ii) is equivalent to the usual Cramer’s condition for the distribution of the jumps of $(S(n))$. It is satisfied if and only if the function

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

is finite in a neighborhood of the set $\tilde{D} \stackrel{\text{def.}}{=} \{(\alpha, \beta) \in \mathbb{R}^2 : P(e^{\alpha}, e^{\beta}) \le 1\}$. Similarly, Assumption (A3) (ii) is satisfied if and only if the generating functions

$$ \tilde{\phi}_i(\alpha, \beta) \stackrel{\text{def.}}{=} \phi_1(e^{\alpha}, e^{\beta}) = \sum_{k=(k_1, k_2)} \mu_i(k) e^{\alpha k_1 + \beta k_2}, \quad i \in \{0, 1, 2\}, $$

are finite in a neighborhood of $\tilde{D}$. Since $(0,0) \in \tilde{D}$, all jumps of the random walk $(Z(n))$ are in particular integrable.

Remark furthermore that because of Assumptions (A1) (ii) and (A3) (iv), the mean jumps

$$ (M_1, M_2) = \sum_{j \in \mathbb{Z}^2} j\mu(j), \quad (M_1^1, M_2^1) = \sum_{j \in \mathbb{Z}^2} j\mu_1(j) \quad \text{and} \quad (M_1^2, M_2^2) = \sum_{j \in \mathbb{Z}^2} j\mu_1(j) $$

are non zero, $M_2^1 > 0$ and $M_1^2 > 0$. Moreover, since with Assumption (A1) (iii), the sets $D \cap D_1$ and $D \cap D_2$ have a non-empty interior, one has also $M_1 M_2^1 \neq M_2 M_1^1$ and $M_2 M_1^2 \neq M_1 M_2^2$.

Under our assumptions, the necessary and sufficient conditions of positive recurrence and transience for the Markov chain $(Z(n))$, are given in the next proposition. See Theorem 3.3.1 of [7] for example.

**Proposition A.1.** Under the assumptions (A1)-(A3) then, the following assertions hold for the Markov chain $(Z(n))$ on $\mathbb{Z}_+^2$.

(1) Positive recurrence. If and only if one of the following conditions is satisfied:

$$ (R0) M_1 < 0, M_2 < 0, M_1 M_2^1 < M_2 M_1^1 \text{ and } M_2 M_1^2 < M_1 M_2^2; $$
(R1) $M_2<0, M_1\ge 0$ and $M_1M_2^1<M_2M_1^1$;

(R2) $M_1<0, M_2\ge 0$ and $M_2M_1^2<M_1M_2^2$.

(2) Transience. If one of the following conditions is satisfied:

(T0) $M_1>0$ and $M_2>0$;

(T1) $M_2<0$ and $M_1M_2^1>M_2M_1^1$;

(T2) $M_1<0$ and $M_2M_1^2>M_1M_2^2$;

The proposition below studies the relation between conditions of transience and positive recurrence for the Markov chain $(Z(n))$ and the location of the points defining the regions (B0)-(B7). We formulate the conditions (R0)-(R2) and (T0)-(T3) in terms of the location of the points $(x^*, Y_i(x^*))$, $(X_i(y^*), y^*)$, and $(x^{**}, Y_i(x^{**}))$, $(X_i(y^{**}), y^{**})$, $i\in\{1, 2\}$. In Proposition A.4, the relation with the location of the dominant singularities $x_d$ and $y_d$ is analyzed.

We first establish a technical lemma.

**Lemma A.1.** Under the assumptions (A1)-(A3), then

(i) $M_2<0$ if and only if $(1, 1)\in S_{11}\cup S_{21}$ and $x_P^*<1<x_P^{**}$.
In this case, $Y_1(1)=1$ and

$$1 = \begin{cases} x^* & \text{if } M_1 M_2^1 < M_2 M_1^1, \\ x^{**} & \text{if } M_1 M_2^1 > M_2 M_1^1. \end{cases} \qquad (A.1)$$

(ii) $M_1<0$ if and only if $(1, 1)\in S_{11}\cup S_{12}$ and $y_P^*<1<y_P^{**}$.
In this case, $X_1(1)=1$ and

$$1 = \begin{cases} y^* & \text{if } M_2 M_1^2 < M_1 M_2^2, \\ y^{**} & \text{if } M_2 M_1^2 > M_1 M_2^2. \end{cases} \qquad (A.2)$$

*Proof.* To prove (i), it is sufficient to remark that $M_2=\partial_y P(1, 1)$ and that, according to the definition of the curves $S_{ij}$, $i, j\in\{1, 2\}$,

$$S_{11} \cup S_{21} = \{(x, y) \in \partial D : \partial_y P(x, y) \le 0\}$$

and that by Lemma 4.1, the points $(x_P^*, Y_1(x_P^*))=(x_P^*, Y_2(x_P^*))$ and $(x_P^{**}, Y_1(x_P^{**}))=(x_P^{**}, Y_1(x_P^{**}))$ are the only points on the boundary $\partial D$ of the set $D$ satisfying the relation $\partial_y P(x, y)=0$.

Moreover, by relations (2.15) and (2.17), we have $S_{11}\cup S_{21}=\{(x, y) \in \partial D : y=Y_1(x)\}$. Hence, when $M_2<0$, one has $Y_1(1)=1$ and therefore $\phi_1(1, Y_1(1))=\phi_1(1, 1)=1$. By Corollary 4.1, it follows that the point 1 is an end point of the line segment $[x^*, x^{**}]$, and, consequently, with relations (2.9) and (2.10), we conclude that one and only one of the following cases occurs:

$$-x_P^* < x^* = 1 < x^{**} \le x_P^{**};$$

$$-x_P^* \le x^* < 1 = x^{**} < x_P^{**}.$$

By the implicit function theorem we have, for any $x\in|x_P^*, x_P^{**}|$,

$$\frac{d}{dx}\phi_1(x, Y_1(x)) = \partial_x\phi_1(x, y) + \partial_y\phi_1(x, y)\partial_x P(x, y)/\partial_y P(x, y)|_{(y=Y_1(x))},$$

hence, when $M_2<0$, the relation

$$\left. \frac{d}{dx} \phi_1(x, Y_1(x)) \right|_{x=1} = \partial_x \phi_1(1, 1) + \partial_y \phi_1(1, 1) \partial_x P(1, 1) / \partial_y P(1, 1) = M_1^1 - M_2^1 M_1 / M_2,$$

holds, and, consequently, in a neighborhood of $x=1$, the function $x\mapsto\phi_1(x, Y_1x)$ is
- non-decreasing if $M_1^1 - M_2^1 M_1 / M_2 > 0$,

- non-increasing if $M_1^1 - M_2^1 M_1 / M_2 < 0$.

By Corollary 4.1, we have the relation $\phi_1(x, Y_1(x)) < 1$, for all $x \in ]x^*, x^{**}[, \text{ and } \phi_1(x, Y_1(x)) > 1$, for $x \in [x_P^*, x^*[\cup]x^{**}, x_P^{**}]$. Hence, when $x_P^* < x^* = 1 < x^{**} \le x_P^{**}$, the function $x \mapsto \phi_1(x, Y_1(x))$ is non-decreasing in a neighborhood of $x=1$ and $M_1^1 M_2 - M_2^1 M_1 > 0$, and, when $x_P^* \le x^* < 1 = x^{**} < x_P^{**}$, the function $x \mapsto \phi_1(x, Y_1(x))$ is non-decreasing in a neighborhood of $x=1$ and $M_1^1 M_2 - M_2^1 M_1 < 0$.

The first assertion of our lemma is proved. The second assertion is symmetrical by exchanging the roles of $x$ and $y$. □

**Proposition A.2.** Under Assumptions (A1)-(A3), then

(i) (R0) holds if and only if $(x^*, Y_1(x^*)) = (X_1(y^*), y^*) = (1, 1)$, $x_P^* < 1$ and $y_P^* < 1$;

(ii) (R1) " $(x^*, Y_1(x^*)) = (1, 1)$ and $x_P^* < 1$;

(iii) (R2) " $(X_1(y^*), y^*) = (1, 1)$ and $y_P^* < 1$.

(iv) (T0) " $(1, 1) \in S_{22}$, $1 < x^{**} < 1 < y^{**}$.
In this case, $Y_2(x^{**}) < 1$ and $X_2(y^{**}) < 1$.

(v) (T1) holds if and only if $(1, 1) = (x^{**}, Y_1(x^{**}))$ and $1 < x_P^{**}$.

(vi) (T2) " $(1, 1) = (X_1(y^{**}), y^{**})$ and $1 < y_P^{**}$.

*Proof.* The assertions (i)-(iii) and (v)-(vi) follow directly from Lemma A.1.

We have only to establish (iv). By using again the relations $M_1 = \partial_x P(1, 1)$ and $M_2 = \partial_y P(1, 1)$, when $(1, 1) \in S_{22}$, by relations (2.9), (2.10) and (2.18), and with Lemma 4.1, we get the identities $(1, 1) = (1, Y_2(1)) = (X_2(1), 1)$ and the relations

$$x_P^* < X_1(y_P^{**}) \le 1 \le x^{**} \le x_P^{**} \quad \text{and} \quad y_P^* < Y_1(x_P^{**}) \le 1 \le y^{**} \le y_P^{**}, \quad (A.3)$$

and

$$M_1 > 0 \Leftrightarrow (1, 1) \neq (X_1(y_P^{**}), y_P^{**}) \Leftrightarrow 1 < y_P^{**}, \quad (A.4)$$

$$M_2 > 0 \Leftrightarrow (1, 1) \neq (x_P^{**}, Y_1(x_P^{**})) \Leftrightarrow 1 < x_P^{**}. \quad (A.5)$$

Hence, when $(1, 1) \in S_{22}$, we have $1 < x^{**}$ and $1 < y^{**}$, the condition (T0) holds.

Conversely, if (T0) holds, then, according to the definition of the curve $S_{22}$,

$$(1, 1) = (1, Y_2(1)) = (X_2(1), 1) \in S_{22} \quad (A.6)$$

and, by using relations (A.4) and (A.5), we get

$$1 < x_P^{**}, \quad \text{and} \quad 1 < y_P^{**}.$$

By relation (A.3), we have always $1 \le x^{**}$ and, if we assume that $x^{**} = 1$, then, by relations (A.6) and (A.3), we obtain $x_P^* < 1 = x^{**} < x_P^{**}$. Consequently, by Lemma 4.1 and Corollary 4.1, we have $Y_1(1) < Y_2(1)$ and $\phi_1(1, Y_1(1)) = 1$.

Since under our assumptions (A3) (ii) and (A3) (v), the function $y \mapsto \phi_1(1, y)$ is strictly increasing on the line segment $[0, Y_1(1)]$, these relations imply that $\phi_2(1, Y_2(1)) > 1$ and, consequently, $(1, Y_1(1)) \neq (1, 1)$. Since this last relation contradicts relation (A.6), we conclude that $x^{**} > 1$.
By exchanging the roles of $x$ and $y$, the same arguments prove also that when (T0) holds, the relation $y^{**}>1$ holds. By relation (2.18), we have

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

and, by Lemma 4.1, the functions

$$X_2 : [Y_1(x_P^{**}), y_P^{**}] \to [X_1(y_P^{**}), x_P^{**}] \text{ and } Y_2 : [X_1(y_P^{**}), x_P^{**}] \to [Y_1(x_P^{**}), y_P^{**}]$$

are strictly decreasing. It follows that $Y_2(x^{**})<Y_2(1)=1$ holds and, similarly, $X_2(y^{**})<X_2(1)=1$. The assertion (iv) of Proposition A.2 is proved. □

The following statement give relations between the conditions (B0)-(B7) and the conditions (R0)-(R2) and (T0)-(T2).

**Proposition A.3.** Under the assumptions (A1)-(A3), then

(1) both conditions (T1) and (T2) hold if and only if (B7) holds;

(2) (B2) holds if (T0) holds.

(3) If either (B3) or (B4) holds, then

- (T2) is not possible;

- (T1) holds if and only if $x^{**}=1$, and in this case, $Y_1(1)=1$;

- either (R0) or (R1) or (R2) holds whenever $x^{**}>1$.

(4) If either (B5) or (B6) holds, then

- (T1) is not possible;

- (T2) holds if and only if $y^{**}=1$, and in this case, $X_1(1)=1$;

- either (R0) or (R1) or (R2) holds whenever $y^{**}>1$.

*Proof.* The first and the second assertions of this statement follow from directly Proposition 2.2 and Proposition A.2.

If either (B3) or (B4) holds, by Proposition 2.2, $1<Y_2(x^{**})\le y^{**}$ holds, and, consequently, by Proposition A.2, the case (T2) is not possible.

Still under the condition that either (B3) or (B4) holds. By Proposition 2.2 and with relations (2.9) and (2.10), we have $1\le x^{**}<x_P^{**}$. Therefore, by Proposition A.2, (T1) holds if and only if $(1,1)=(x^{**}, Y_1(x^{**}))$ holds. Since $P(1,1)=1$, by Lemma 4.1, either $Y_1(1)=1$ or $Y_2(1)=1$ holds, since, in our case, we have $1<Y_2(x^{**})$ by Proposition 2.2, it follows that when either (B3) or (B4) holds, the condition (T1) is satisfied if and only if $x^{**}=1$.

Under the condition that either (B3) or (B4) holds and that $x^{**}>1$ holds. We have that either $M_2\le 0$ or $M_1\le 0$, because otherwise (T0) and, consequently, (B2) holds. By Proposition 2.2 and with relation (2.9), we get the relations $1<Y_2(x^{**})\le y^{**}\le y_P^{**}$ and $1<x^{**}\le x_P^{**}$, and, consequently, $(1,1)\ne(x_P^{**}, Y_1(x_P^{**}))$ and $(1,1)\ne(X_1(y_P^{**}), y_P^{**})$. Since the point $(x,y) = (x_P^{**}, Y_1(x_P^{**}))$, resp. $(x,y)=(X_1(y_P^{**}), y_P^{**})$) is the only point on the boundary $\partial D$ of D for which $\partial_x P(x,y) > 0$ and $\partial_y P(x,y) = 0$, resp. $\partial_x P(x,y)=0$ and $\partial_y P(x,y)>0$) and, since $M_1=\partial_x P(1,1)$ and $M_2=\partial_y P(1,1)$, we conclude therefore that, either $M_1<0$ or $M_2<0$. If $M_2<0$, then, by Lemma A.1, and the assumption $x^{**}>1$, we get $x^*=1$ and $M_1 M_2^1 < M_2 M_1^1$. Similarly, if $M_1<0$, by Lemma A.1 and since, by Proposition 2.2, $1<Y_2(x^{**})\le y^{**}$ holds, we obtain $y^*=1$ and $M_2 M_1^2 < M_1 M_2^2$. We conclude that in the case when either (B3) or (B4) holds and $x^{**}>1$, one of the conditions (R0), (R1) or (R2) is satisfied.

The third assertion of Proposition A.3 is therefore also proved. The last assertion follows by exchanging the roles of $x$ and $y$. □
We can now establish the relations with the locations of the dominant singularities $x_d$ and $y_d$.

**Proposition A.4.** Under the assumptions (A1)-(A3) the following assertions hold:

(i) If the Markov chain $(Z(n))$ is positive recurrent then $x_d>1$ and $y_d>1$.

(ii) If (T0) is satisfied then (B2), $x_d=x^{**}>1$ and $y_d=y^{**}>1$ hold.

(iii) If (T1) is satisfied then

- in the cases (B0)-(B2), relations $x_d=x^{**}=1<x_P^{**}$ and $y_d=y^{**}>Y_1(1)=1$ hold;

- in the cases (B3) and (B4), relations $x_d=x^{**}=1<x_P^{**}$ and $y_d=Y_2(1)>1$ holds;

- (B5) and (B6) do not hold.

(iv) If (T2) is satisfied then

- in the cases (B0)-(B2), relations $y_d=y^{**}=1<y_P^{**}$ and $x_d=x^{**}>X_1(1)=1$ hold;

- and (B3) and (B4) do not hold;

- in the cases (B5) and (B6), relations $y_d=y^{**}=1<y_P^{**}$ and $x_d=X_2(1)>1$ hold.

*Proof.* Consider first the case when one of the cases (B0)-(B2) occurs. Then by Proposition 2.2 and according to the definition of $x_d$ and $y_d$,

$$x_d = x^{**} > X_1(y^{**}) \quad \text{and} \quad y_d = y^{**} > Y_1(x^{**}). \qquad (A.7)$$

Hence, if we suppose moreover that (R0) holds, by Lemma A.1 and using (2.9), we obtain

$$1 = x^* < x^{**} = x_d \quad \text{and} \quad 1 = y^* < y^{**} = y_d.$$

If we suppose that (R1) holds, then by Proposition A.2 and using (2.9), and (A.7),

$$1 = x^* < x^{**} = x_d, \quad \text{and} \quad 1 = Y_1(x^*) < y_d.$$

And similarly, if we suppose that (R2) holds, then by Proposition A.2 and using (2.9), and (A.7),

$$1 = y^* < y^{**} = y_d, \quad \text{and} \quad 1 = X_1(y^*) < x_d.$$

In the case when one of the cases (B0)-(B2) holds, the first assertion of our statement is therefore proved.

Suppose now that either (B3) or (B4) holds. In this case, by Proposition 2.2 and according to the definition of $x_d$ and $y_d$, we have

$$x_d = x^{**} < x_P^{**} \quad \text{and} \quad y_d = Y_2(x^{**}) > 1. \qquad (A.8)$$

Moreover, by relation (2.10), the inequality $1 \le x^{**}$ always holds, and consequently, in this case, by Proposition A.3, one of the conditions (R0)-(R2) is satisfied if and only if $1 < x^{**} = x_d$. Hence, when either (B3) or (B4) holds, the first assertion of Proposition A.4 is also proved. The same arguments (it is sufficient to exchange the roles of $x$ and $y$) prove the first assertion of our proposition in the case when either (B5) or (B6) holds.

Furthermore, if the condition (T0) is satisfied, then by Proposition A.3, (B2) holds and consequently, by Proposition A.2 and using (A.7) one gets $x_d = x^{**} > 1$ and $y_d = y^{**} > 1$. The second assertion of Proposition A.4 is therefore also proved.
Suppose now that the condition (T1) is satisfied. Then by Proposition A.3, the cases (B5) and (B6) are impossible, and by Proposition A.2,

$$x^{**} = 1 \quad \text{and} \quad Y_1(x^{**}) = 1. \qquad (A.9)$$

Moreover, if one of the cases (B0)-(B2) holds, then by (A.7), one gets $y_d = y^{**} > Y_1(x^{**}) = 1$, and if either (B3) or (B4) holds, then by (A.8) and using the second relation of (A.9), we obtain $y_d = Y_2(x^{**}) > Y_1(x^{**}) = 1$. The third assertion of Proposition A.4 is therefore also proved. The proof of fourth assertion is the same, it is sufficient to exchange the roles of $x$ and $y$. □

# B Irreducibility Properties of the killed Markov Chain

## Proof of Lemma 2.2

Since by the assumption (A1), the homogeneous random walk $(S(n))$ with transition probabilities $\mathbb{P}_j(S(1) = k) = \mu(k-j)$ is irreducible on $\mathbb{Z}^2$, for any $k \in \mathbb{Z}^2$, there is a sequence of points $\ell^k(1) = (\ell_1^k(1), \ell_2^k(1)), \dots, \ell^k(N_k) = (\ell_1^k(N_k), \ell_2^k(N_k)) \in \mathbb{Z}^2$ such that $\ell^k(1) + \dots + \ell^k(N_k) = k$ and $\mathbb{P}_0(S(N_k) = k) \ge \mu(\ell_1^k) \times \dots \times \mu(\ell^k(N_k)) > 0$. Let $N_1 = 1+\max\{N_{(-1,0)}, N_{(1,0)}, N_{(0,-1)}, N_{(0,1)}\}$. Then because of Assumption (A2), for any $j = (j_1, j_2) \in \mathbb{Z}_+^2$ with $j_1, j_2 \ge N_1$ and $k \in \{(-1,0), (1,0), (0,-1), (0,1)\}$ the sequence of points $m^k(1) = j + \ell^k(1), \dots, m^k(N_k) = j + \ell^k(1) + \dots + \ell^k(N_k)$ does not exit from the set $(\mathbb{N}^*)^2$, and consequently

$$\mathbb{P}_j(Z(n) = j + k, \text{ for some } n < \tau) \ge \mathbb{P}_j(Z(n) = m^k(n), \quad \forall n \in \{1, \dots, N_k\} > 0.$$

This proves that

$$g(j, k) \ge \mathbb{P}_j(Z(n) = k, \text{ for some } n < \tau) > 0 \quad \text{whenever} \quad j_1, j_2, k_1, k_2 \ge N_1. \qquad (B.1)$$

Consider now a random walk $(\hat{S}(n))$ on $\mathbb{Z} \times \mathbb{N}$ with transition probabilities

$$\mathbb{P}_j(\hat{S}(1) = k) = \begin{cases} \mu(k-j) & \text{for all } j = (j_1, j_2), k \in \mathbb{Z} \times \mathbb{N} \text{ with } j_2 > 0, \\ \mu_1(k-j) & \text{for all } j = (j_1, j_2), k \in \mathbb{Z} \times \mathbb{N} \text{ with } j_2 = 0. \end{cases}$$

By Assumptions (A1) and (A3) (v) and (vi), such a random walk is irreducible on $\mathbb{Z} \times \mathbb{N}$, and consequently, for any $j_2 \in \mathbb{N}$ and $k = (k_1, k_2) \in \mathbb{Z}^2$ such that $(k_1, k_2 + j_2) \in \mathbb{Z} \times \mathbb{N}$, there is a sequence of points $\ell^{j_2,k}(1) = (\ell_1^{j_2,k}(1), \ell_2^{j_2,k}(1)), \dots, \ell^{j_2,k}(N_{j_2,k}) = (\ell_1^{j_2,k}(N_{j_2,k}), \ell_2^{j_2,k}(N_{j_2,k})) \in \mathbb{Z}^2$ such that for any $n \in \{1, \dots, N_{j_2,k}\}$,

$$(0, j_2) + \ell^{j_2,k}(1) + \dots + \ell^{j_2,k}(n) \in \mathbb{Z} \times \mathbb{N},$$

and

$$\mathbb{P}_{(0,j_2)}(\hat{S}(N_{j_2,k}) = (0, j_2) + k) = \mathbb{P}_{(0,j_2)}(\hat{S}(n) - \hat{S}(n-1) = \ell^{j_2,k}(n), \forall n \in \{1, \dots, N_{j_2,k}\}) > 0.$$

Letting

$$N_2 = 1 + \max\{N_{j_2,k} : j_2 \le N_1, (k_1, k_2) \in \{(-1,0), (1,0), (0,-1), (0,1)\}, (k_1, k_2 + j_2) \in \mathbb{Z} \times \mathbb{N}\}$$

and using Assumptions (A2) and (A3) (iv), we obtain that for any $j = (j_1, j_2) \in \mathbb{Z}_+^2$ with $j_1 \ge N_2$, and any $k = (k_1, k_2) \in \{(-1,0), (1,0), (0,-1), (0,1)\}$ such that $j + k \in \mathbb{Z} \times \mathbb{N}$, the sequence of points $m^{j,k}(1) = j + \ell^{j,k}(1), \dots, m^{j,k}(N_{j,k}) = j + \ell^{j,k}(1) + \dots + \ell^{j,k}(N_{j,k}) = (\ell_1^{j,k}(N_{j,k}), \ell_2^{j,k}(N_{j,k}))$ does not exist from the set $\mathbb{N}^* \times \mathbb{N}$, and consequently,

$$\mathbb{P}_j(Z(n) = j + k \text{ for some } n < \tau) \ge \mathbb{P}_j(\hat{S}(n) = m^{j_2,k}(n), \forall n \in \{1, \dots, N_{j_2,k}\}) > 0.$$
This proves that

$$g(j, k) > 0 \quad \text{whenever} \quad j_2 \le N_1, k_2 \le N_1 + 1 \text{ and } j_1, k_1 \ge N_2, \qquad (B.2)$$

and with exactly the same arguments (it is sufficient to exchange the roles of the first and the second coordinates of the points in $Z_+^2$), one gets that for some $N_3 > 0$,

$$g(j, k) > 0 \quad \text{whenever} \quad j_1 \le N_1, k_1 \le N_1 + 1 \text{ and } j_2, k_2 \ge N_3. \qquad (B.3)$$

When combined together, relations (B.1), (B.2) and (B.3) show that for some $N_0 \ge 0$,

$$g(j, k) > 0 \quad \text{for all} \quad j, k \in Z_+^2 \text{ with } \|j\| \ge N_0 \text{ and } \|k\| \ge N_0. \qquad (B.4)$$

Remark now that if $\ell \in Z_+^2$ is such that $g(\ell, j_0) > 0$ for some $j_0 \in Z_+^2$ with $\|j_0\| \ge N_0$, then also $\mathbb{P}_\ell(Z(n) = j_0, \tau > n) > 0$ for some $n \in \mathbb{N}$, and consequently, by (B.4) and using the inequality

$$g(\ell, k) \ge \sum_{j \in Z_+^2 \setminus \{0\}} \mathbb{P}_\ell(Z(n) = j, \tau > n)g(j, k) \ge \mathbb{P}_\ell(Z(n) = j_0, \tau > n)g(j_0, k) \qquad (B.5)$$

we obtain

$$g(\ell, k) \ge \mathbb{P}_\ell(Z(n) = j, \tau > n)g(j, k) > 0 \quad \text{for all } k \in Z_+^2 \text{ with } \|k\| \ge N_0.$$

Hence, for any $\ell \in Z_+^2$ we have either $g(\ell, k) > 0$ for all $k \in Z_+^2$ with $\|k\| \ge N_0$, or $g(\ell, k) = 0$ also for all $k \in Z_+^2$ with $\|k\| \ge N_0$, and in the last case, because of (B.4), $\|\ell\| < N_0$. Letting therefore $E_0 = \{\ell \in Z_+^2 : g(\ell, k) = 0 \text{ for all } k \in Z_+^2 \text{ with } \|k\| \ge N_0\}$ we obtain a finite subset of $Z_+^2$ satisfying (2.32). Remark moreover that this set satisfies also (2.31) because if suppose that $g(\ell, j) > 0$ for some $\ell \in E_0$ and $j \in Z_+^2 \setminus E_0$, then using again the inequality (B.5) and the same arguments as above we would get $g(\ell, k) > 0$ for all $k \in Z_+^2$ with $\|k\| \ge N_0$. Now, to complete the proof of our lemma, it is sufficient to notice that $g(0, k) > 0$ for any $k \in Z_+^2 \setminus \{0\}$, because under our assumptions, the random walk $(Z(n))$ is irreducible on $Z_+^2$, and consequently, the point $0 = (0, 0)$ does not belong to the set $E_0$.
C Figures

<figure><img src="image_6.png" /><figcaption>Figure 1:</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 2:</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 3:</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 4:</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 5: Possible cases for the location of the line segments $[a, b] = [(X_1(y^{**}), y^{**}), (X_2(y^{**}), y^{**})]$ and $[cd] = [(x^{**}, Y_1(x^{**})), (x^{**}, Y_2(x^{**}))]$</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure 6: All possible configurations for the line segments $[a, b] = [(X_1(y^{**}), y^{**}), (X_2(y^{**}), y^{**})]$ and $[cd] = [(x^{**}, Y_1(x^{**})), (x^{**}, Y_2(x^{**}))]$ and the trace of the set $\{(x, y) \in \mathcal{D} : |x| < x_d, |y| < y_d\}$. Cases (B0)-(B6).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 7: Case (B2) with $x_d = x^{**} = x_P^{**}$. In this case, the line segment $[(x^{**}, Y_1(x^{**})), (x^{**}, Y_2(x^{**}))]$ is a single point and $y_d = y^{**} \le y_P^{**}$.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure 8: Case (B5) with $x_d = x^{**} = x_P^{**}$. In this case, the line segment $[(x^{**}, Y_1(x^{**})), (x^{**}, Y_2(x^{**}))]$ is a single point and $y_d = y^{**} = Y_1(x_d) < y_P^{**}$.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 9: Case (B2) with $x_d = x^{**} = x_P^{**}$ and $y_d = y^{**} = Y_P^{**}$. Here, each of the line segments $[(x^{**}, Y_1(x^{**})), (x^{**}, Y_2(x^{**}))]$ and $[(X_1(y^{**}), y^{**}), (X_2(y^{**}), y^{**})]$ is a single point.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 10: Set $\Theta$, cases (B0)-(B2).</figcaption></figure>