# Sharp and Exact Tail Estimates for Multitype Poissonian Galton Watson Processes and Inhomogeneous Cluster Processes

Théo Leblanc

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Théo Leblanc. Sharp and Exact Tail Estimates for Multitype Poissonian Galton Watson Processes and Inhomogeneous Cluster Processes. 2025. hal-05158013

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# Sharp and Exact Tail Estimates for Multitype Poissonian Galton Watson Processes and Inhomogeneous Cluster Processes

Théo Leblanc

CEREMADE, Université Paris Dauphine - PSL

E-mail address: leblanc@ceremade.dauphine.fr

**Abstract.** We prove exponential moments for linear combinations of the number of individuals of each type of a whole multitype Poissonian Galton Watson process. We give sharp estimates for such quantities, which depend on the expectation of the Poissonian offspring distribution and the linear combination. We apply this result to derive type specific exponential moments for multitype Hawkes processes, improving existing results when the dimension is large. We also prove exponential estimates for tails of inhomogeneous Poissonian clusters. These estimates depend on the decay of the interaction functions defining the clusters and are starting points to derive probabilistic properties on branching processes such as Hawkes processes.

## 1. Introduction

Galton Watson processes are one of the simplest examples of branching processes. They are starting points for many more complex processes. Galton Watson processes describe a whole family coming from a single ancestor. The key property of Galton Watson processes is that each individual reproduces independently of the others and has a random number of children according to a common offspring distribution, see [1, 7]. In the present article we focus on the case where the offspring distribution is a Poisson distribution. This particular case remains quite general and fits the case of Hawkes processes, which have applications in biology, epidemiology, finance, neuroscience, earthquake modeling to name but a few, see for example [8, 12, 10, 2]. The link between Poissonian Galton Watson processes and Hawkes processes lies in Poissonian clusters. Hawkes processes admit a cluster decomposition, where clusters arrive according to a Poisson process and each cluster consists in a Galton Watson tree embedded in the real line, see [9]. In this paper we derive estimates for the Laplace transform and for the tails of multitype Galton Watson processes and clusters. The branching properties of such objects lead to fix point equations for the Laplace transforms. In [16] the author studied these fixed point equations for general Galton Watson processes. Restricting ourselves to Poissonian processes allows us to derive sharper and more specific bounds.

For Hawkes processes, the usual way to derive exponential moments for the number of points in a compact interval is to use cluster decomposition and sum all the contributions of each cluster. In previous work, see for example [13] and [6], decay assumptions on the interaction functions were used to ensure that the sum of the contributions of each cluster converges. In [11] it was shown that in fact, the sum of the contributions only depends on the matrix $H$ gathering the $L^1$ norms of the interaction functions, regardless of the decay of these functions. More precisely, the only

<footnote n="1991">1991 Mathematics Subject Classification. 60J80, 60E15, 60G55, 60J85, 47J26.</footnote>

<footnote>Key words and phrases. Galton Watson process, Poisson Clusters, Exponential moments, Hawkes process, Multitype, Branching process, Fixed point equations.</footnote>
relevant assumption is $SpR(H) < 1$, where SpR stands for the spectral radius, the weakest possible assumption since it is required for the process to exist in a stationary version, [4]. An important result established by [11] is the derivation of exponential moments for a whole multitype Galton Watson with offspring distribution $Pois(H)$. Even though the bound in [11] is quantitative, it quickly becomes unadapted in an high dimensional framework. Indeed, when the number of types (for example the number of neurons when a Hawkes process models brain activity) is large, one is typically interested in the number of points coming from one or from a small number of types. The bound in [11] becomes less significant since it is a bound for all the types at once and depends on the total number of types which could go to infinity.

In this paper we derive exponential estimates for Poissonian Galton Watson processes that are type specific, quantitative and stated under the optimal assumption, i.e. $SpR(H) < 1$. This result applies to Hawkes processes, leading to quantitative and type specific exponential estimates which drastically improve the global bound proved in [11] when the dimension is large. In this work we also derive tail estimates for Galton Watson processes in terms of the generations of the individuals. More precisely, we provide sharp and non asymptotic estimates for the exponential moment of individuals whose generation is at least $n$ with $n \to \infty$. This result is also stated in a type specific framework where the information about types is kept.

We then study structural, topological and exact properties of the Laplace transform of Poissonian Galton Watson processes, leading for example to a precise description of the set of vectors for which the exponential moments are finite.

Lastly we provide tail estimates for inhomogeneous Poissonian clusters in terms of the temporal embedding of the clusters, also in a type specific framework. Temporal tail estimates, for clusters, are derived in the following three cases. First when the interaction functions are exponentially decaying, second when the decay follows a power law, improving previous results where compactly supported interaction functions where considered in the single type case, [13]. And lastly when the interactions functions have bounded support where we obtains the better bound by taking advantage of this strong assumption. Estimates for the tail of clusters are the starting point for many probabilistic results for Hawkes processes such as mixing properties, see [3], or regenerative properties, see [5]. The main originality of this work is that it goes beyond the homogeneous case: the results for Poissonian clusters are stated for inhomogeneous clusters, meaning that the interactions functions may depend on both $s \in \mathbb{R}$ and $t \in \mathbb{R}$ (non convolutive framework) and not just on the delay $t-s$ (convolutive framework) between the current time $t$ and the time $s$ of the previous individual. Similar cases were considered by [14] where the authors also studied properties of the Laplace transform. The non convolutive framework allows more flexibility and improves the possibilities for applications in neurosciences, finance etc.

The present paper is structured as follows. In Section 2 we introduce the objects and notations used in this paper. In Section 3 we present sharp bounds on the exponential moments of the number of points of given types in a Poissonian Galton Watson process. Then we show how exponential moments for Poissonian Galton Watson processes applies to Hawkes processes. Finally we also present results about tail estimates in terms of generations of the individuals for Poissonian Galton Watson processes. In Section 4 we state tail estimates for Poissonian cluster processes. In Section 5 we perform the exact study of the topological properties of the Laplace transform of Galton Watson processes. Technical results on bivariate convolution are needed in Section 4, they are presented and proved in Appendix A. Finally, the proofs can be found in Section 6.
## 2. Settings

Let $\mathbb{N}$ the set of non negative integers and $\mathbb{N}^* = \mathbb{N} \setminus \{0\}$. Let $\mathbb{R}$ the set of real numbers and $\mathbb{R}_+ = [0, \infty)$ the set of non negative real numbers. Consider $M \in \mathbb{N}^*$ and denote by $M$ the set $[1, M]$.

When, in the sequel, Poissonian Galton Watson processes are considered, in Sections 3 and 5, the matrix $H$ is a matrix with non negative coefficients, $H = (H_{ij})_{i,j \in M}$, corresponding to the expectation of the offspring distribution of the Galton Watson process.

When cluster processes are consider, in Section 4, there distribution is characterized by bivariate functions $h = (h_{ij})_{i,j \in M}$ where $h_{ij} : \mathbb{R} \times \mathbb{R} \to \mathbb{R}_+$ are measurable functions. The matrix $H$ stands $H = (\sup_{s \in \mathbb{R}} \|h_{ij}(s, \cdot)\|_{L^1})_{i,j \in M}$.

In this paper we use Ulam Harris Neveu to label individuals of trees. A tree $\tau$ is therefore a subset of $\cup_{n \in \mathbb{N}} (\mathbb{N}^*)^n$ with $(\mathbb{N}^*)^0 = \{\emptyset\}$ such that for any $a, b \in \tau$, individual $b$ is a child of $a$ if and only if $b = ak$ with $k \in \mathbb{N}^*$.

An $M$-type tree, meaning a tree with types in $M$, is an object $(a, \text{tp}(a))_{a \in \tau}$ where $\tau$ is a tree and $\text{tp}(a) \in M$ is the type of $a$. A random tree (resp. $M$-type tree) is a random variable taking values that are trees (resp. $M$-type trees).

### 2.1. $M$-type Poissonian Galton Watson processes.
Let us introduce $M$-type Galton Watson processes with offspring measure $\mathcal{P}ois(H)$. Let a type $m_0 \in M$. The random $M$-type tree $\mathcal{T}^{m_0} = (a, \text{tp}(a))_{a \in \mathcal{A}}$ is an $M$-type Galton Watson process with root of type $m_0$ and offspring measure $\mathcal{P}ois(H)$ if

* $\mathcal{A}$ is a random tree with root $\emptyset$ and $\text{tp}(\emptyset) = m_0$.

* Each $a \in \mathcal{A}$ with type $\text{tp}(a) = m$ reproduces as follows: independently for each $m' \in M$, it has $\mathcal{P}ois(H_{m,m'})$ children of type $m'$.

For a random $M$-type tree $\mathcal{T} = (a, \text{tp}(a))_{a \in \mathcal{A}}$ we define its $M$-type cardinal, a vector of $\mathbb{R}^M$, by

$$ \text{Card}_{\mathcal{M}}(\mathcal{T}) = (\text{Card}(\{a \in \mathcal{A} \mid \text{tp}(a) = m\}))_{m \in \mathcal{M}}. $$

For $a \in \mathcal{A}$, by the Ulam Harris Neveu notation, there exists $n \in \mathbb{N}$ such that $a \in (\mathbb{N}^*)^n$. The integer $n$ is called the generation on $a$ and is denoted $|a|$. For $n \in \mathbb{N}$ we define

$$ \mathcal{T}_n = (a, \text{tp}(a))_{a \in \mathcal{A}, |a|=n}, \quad \mathcal{T}_{\le n} = (a, \text{tp}(a))_{a \in \mathcal{A}, |a| \le n}, \quad \mathcal{T}_{\ge n} = (a, \text{tp}(a))_{a \in \mathcal{A}, |a| \ge n}. $$

We also have similar definitions for the $M$-type cardinal of $\mathcal{T}_n$, $\mathcal{T}_{\le n}$ and $\mathcal{T}_{\ge n}$.

### 2.2. Inhomogeneous Poissonian clusters.
$M$-type Galton Watson processes are closely linked to linear multivariate Hawkes processes and more generally to Poissonian clusters.

A random variable $G_{t_0}^{m_0} = (a, \text{tp}(a), \text{bd}(a))_{a \in \mathcal{A}}$ where $\mathcal{A}$ is a random tree, $\text{tp}(a) \in M$ is the type of $a$ and $\text{bd}(a) \in \mathbb{R}$ is the birth date of $a$, is a $\mathcal{P}ois(h)$ inhomogeneous $M$-type cluster with root of type $m_0$ and born at time $t_0$ if:

* The root of $\mathcal{A}$ is $\emptyset$ with $\text{tp}(\emptyset) = m_0$ and $\text{bd}(\emptyset) = t_0$.

* Each $a \in \mathcal{A}$ with type $\text{tp}(a) = m \in M$ reproduces as follows:

- independently, for all $m' \in M$, $a$ has $\mathcal{P}ois(\|h_{m,m'}(\text{bd}(a), \cdot)\|_{L^1})$ children of type $m'$,
– independently, for each b child of a with type tp(b) = m' we have the density

$$ \mathrm{bd}(b) \sim \frac{h_{m,m'}(\mathrm{bd}(a), t)}{\|h_{m,m'}(\mathrm{bd}(a), \cdot)\|_{L^1}} dt. $$

*Remark 2.1.* If for any $i, j \in \mathcal{M}$ we have that $h_{ij}(s, t)$ only depends on $t-s$ we are in the homogeneous case. With a slight abuse of notation, we still call $h_{ij}$ the function that satisfies

$$ h_{ij}(s, t) = h_{ij}(t-s). $$

If moreover one takes $h_{ij}(s, t) = h_{ij}(t-s)$ such that $h_{ij}(s, t) = 0$ for $t < s$, one recovers the usual clusters of Hawkes processes, see [8] and [9]. Moreover, in this case, if one omits the birth dates, one obtains an $\mathcal{M}$-type Galton Watson process with offspring measure $\mathcal{P}ois(H)$.

Given $I \subset \mathbb{R}$ and a cluster $G = (a, \mathrm{tp}(a), \mathrm{bd}(a))_{a \in \mathcal{A}}$ we define the trace of $G$ on $I$ by

$$ G \cap I = (a, \mathrm{tp}(a), \mathrm{bd}(a))_{a \in \mathcal{A}, \mathrm{bd}(a) \in I}. $$

Finally we also define the number of individuals of $G$ with birth date in $I$ by

$$ \mathrm{Card}(G \cap I) = \mathrm{Card}(\{a \in \mathcal{A} \mid \mathrm{bd}(a) \in I\}) $$

and the typed cardinal

$$ \mathrm{Card}_{\mathcal{M}}(G \cap I) = \left( \mathrm{Card}(\{a \in \mathcal{A} \mid \mathrm{bd}(a) \in I, \mathrm{tp}(a) = m\}) \right)_{m \in \mathcal{M}}. $$

### 2.3. Notations.
Along this paper we use the following notations.

For any vector $x \in \mathbb{R}^M$ we denote $|x|_\infty$ the norm on $\mathbb{R}^M$ that calculates the maximum absolute value of the coordinates of $x$. In other words,

$$ |x|_\infty = \max_{m \in \mathcal{M}} |x_m|. $$

We denote by $\mathbb{R}_+^M$ the vector of $\mathbb{R}^M$ with nonnegative coordinates,

$$ \mathbb{R}_+^M = \{x \in \mathbb{R}^M \mid \forall i \in \mathcal{M}, x_i \ge 0\}. $$

The operator norm associated to $|\cdot|_\infty$ is defined as follows. For $A$ a matrix of size $n \times p$ we have

$$ \|A\|_{\infty} = \sup_{x \in \mathbb{R}^p \setminus \{0\}} \frac{|Ax|_{\infty}}{|x|_{\infty}}. $$

For $m \in \mathcal{M}$, the vector $e_m \in \mathbb{R}^M$ is the vector with zeros everywhere except at coordinate $m$ where it is one,

$$ e_m = (\mathbb{1}_{m'=m})_{m' \in \mathcal{M}}, $$

and $\mathbf{1} = (1, \dots, 1)^T \in \mathbb{R}^M$.

The usual scalar product between two vectors $x = (x_m)_{m \in \mathcal{M}} \in \mathbb{R}^M$ and $y = (y_m)_{m \in \mathcal{M}} \in \mathbb{R}^M$ is denoted by $x \cdot y$,

$$ x \cdot y = \sum_{m \in \mathcal{M}} x_m y_m. $$

For any real function $f: \mathbb{R} \to \mathbb{R}$ and any vector $x \in \mathbb{R}^M$, we define $f(x)$ by the coordinatewise application of $f$,

$$ f(x) = (f(x_m))_{m \in \mathcal{M}}. $$

For any vector $x$ we define $\mathrm{diag}(x)$ the diagonal square matrix, its dimension is given by the dimension of $x$, with diagonal coefficients given by $x$.
We denote by $M_{n,p}(\mathbb{R})$ the set of real matrices of size $n \times p$. For matrices $A$ and $B$ of $M_{n,p}(\mathbb{R})$ we define the partial orders $\prec$ and $\preceq$ by:

$$A \prec B \Leftrightarrow \forall i, j, A_{ij} < B_{ij}$$

and

$$A \preceq B \Leftrightarrow \forall i, j, A_{ij} \leq B_{ij}.$$

The spectral radius of a real square matrix $A$ is denoted by $\text{SpR}(A)$.

For any matrix $B = (B_{ij})_{i,j}$ the matrix $|B|$ is defined by

$$|B| = (|B_{ij}|)_{i,j}.$$

The $L^1$ norm of a real function $f: \mathbb{R} \to \mathbb{R}$ is defined by

$$\|f\|_{L^1} = \int_{\mathbb{R}} |f(x)|dx.$$

The $L^\infty$ norm of $f$ is defined by

$$\|f\|_{L^\infty} = \sup_{x \in \mathbb{R}} |f(x)|.$$

For a matrix function $z: \mathbb{R} \to M_{n,p}(\mathbb{R})$ we define its $L^1$ and $L^\infty$ norms by gathering the coordinatewise norms,

$$\|z\|_{L^1} = (\|z_{ij}\|_{L^1})_{i,j}, \quad \|z\|_{L^\infty} = (\|z_{ij}\|_{L^\infty})_{i,j}.$$

For $v: \mathbb{R}^2 \to M_{n,p}(\mathbb{R})$ and $w: \mathbb{R}^2 \to M_{p,q}(\mathbb{R})$ measurable bivariate matrix functions, we define their bivariate convolution $v \star w: \mathbb{R}^2 \to M_{n,q}(\mathbb{R})$, when defined, by

$$v \star w(s, t) = \int_{-\infty}^{\infty} v(s, x) \times w(x, t)dx$$

where $\times$ stands for the matricial product.

*Remark 2.2.* If one takes $v(s, t) = f(t-s)$ and $w(s, t) = g(t-s)$ then one has $v \star w(s, t) = f \star g(t-s)$ the usual convolution for univariate functions.

We also define the residual integral $R_v: \mathbb{R}^2 \to M_{n,p}(\mathbb{R})$ by

$$R_v(s, t) = \int_t^\infty v(s, x)dx,$$

and the uniform residual integral

$$R_v^\infty(d) = \|s \in \mathbb{R} \mapsto R_v(s, s+d)\|_{L^\infty}.$$

*Remark 2.3.* One has that $R_v = v \star (1_{D_-} \text{Id})$ where $D_- = \{(s, t) | t \leq s\}$.

For a bivariate matrix function $v$ we define

$$\|v\|_{L^\infty L^\infty} = \|s \in \mathbb{R} \mapsto \|v(s, \cdot)\|_{L^\infty}\|_{L^\infty} = \left( \sup_{s,t \in \mathbb{R}} |v_{ij}(s,t)| \right)_{i,j}$$

and

$$\|v\|_{L^\infty L^1} = \|s \in \mathbb{R} \mapsto \|v(s, \cdot)\|_{L^1}\|_{L^\infty} = \left( \sup_{s \in \mathbb{R}} \|v_{ij}(s, \cdot)\|_{L^1} \right)_{i,j}.$$

Finally, let us introduce the following definition.
**Definition 2.4.** Let $A$ a real square matrix. We say that $A$ satisfies the property $\mathbf{Ge}(r, K)$ with $r, K \ge 0$ and denote $A \in \mathbf{Ge}(r, K)$ if

$$ \forall n \in \mathbb{N}^*, \|A^n\|_{\infty} \le Kr^n. $$

*Remark 2.5.* The well known Gelfand’s Theorem states that for any square matrix $A$, for any norm $\| \cdot \|$, we have $\|A^n\|^{1/n} \xrightarrow{n \to \infty} \text{SpR}(A)$. Thus, for any $r > \text{SpR}(A)$ there exists $K < \infty$ such that $A \in \mathbf{Ge}(r, K)$ and we have $\text{SpR}(H) < 1 \iff \exists r < 1, \exists K < \infty, H \in \mathbf{Ge}(r, K)$. Since we are interested in subcritical Galton Watson processes and clusters, the well-known condition $\text{SpR}(H) < 1$ can be replaced by $H \in \mathbf{Ge}(r, K)$ for some $r < 1$ and $K < \infty$.

### 3. Main results for Galton Watson processes and application to Hawkes processes

In this section, we state three main results about exponential estimates for Galton Watson processes and the applications to Hawkes processes. The proofs of these theorems are based on fix point methods.

**3.1. Type specific exponential estimates for Galton Watson processes.** Theorem 3.1 gives sharp bounds on the exponential moments of the scalar product between a linear combination $u \in \mathbb{R}_+^M$ and the $M$-type cardinal of a $\mathcal{P}$ois($H$) Galton Watson processes. It also introduces the function $u \mapsto \mathcal{L}(u)$ whose domain is the focus of Section 5. Theorem 3.1 is proved in Section 6.1.

**Theorem 3.1.** Let $\mathcal{T}^m$ be a $\mathcal{P}$ois($H$) Galton Watson processes with root of type $m \in \mathcal{M}$. For all $u \in \mathbb{R}_+^M$ define $\mathcal{L}(u) \in [0, \infty]^M$ by,

$$ \exp (e_m \cdot \mathcal{L}(u)) = \mathbb{E} \left[ e^{u \cdot \text{Card}_M(\mathcal{T}^m)} \right], \quad \forall m \in \mathcal{M}. \quad (3.1) $$

Then $\mathcal{L}(u) \in [0, \infty]^M$ satisfies the following properties.

(1) We have the following finiteness condition,

$$ |\mathcal{L}(u)|_{\infty} < \infty \iff \exists x \in \mathbb{R}_+^M, x = u + H(e^x - 1). \quad (3.2) $$

(2) If $|\mathcal{L}(u)|_{\infty} < \infty$ then we have

$$ \mathcal{L}(u) = u + H(e^{\mathcal{L}(u)} - 1) \quad (3.3) $$

and $\mathcal{L}(u)$ is the smallest solution (for $\le$) of Equation (3.3) among the solutions $y \in \mathbb{R}_+^M$.

(3) For any $u, v \in \mathbb{R}_+^M$ and $0 \le s \le 1$ we have

$$ \mathcal{L}(u) + \mathcal{L}(v) \le \mathcal{L}(u+v), \quad \text{and} \quad \mathcal{L}(su + (1-s)v) \le s\mathcal{L}(u) + (1-s)\mathcal{L}(v). \quad (3.4) $$

(4) For $u \in \mathbb{R}_+^M$, if we have $\text{SpR}(H \text{diag}(e^{\mathcal{L}(u)})) < 1$ then we have

$$ \mathcal{L}(u) \le (\text{Id} - H \text{diag}(e^{\mathcal{L}(u)}))^{-1} u. \quad (3.5) $$

(5) If $H \in \mathbf{Ge}(r, K)$ with $r < 1$ and $K < \infty$, and if we define

$$ t_0(r, K) = \frac{\log\left(\frac{1+r}{2r}\right)}{1 + \frac{2K}{1-r}}, \quad (3.6) $$

for all $u \in \mathbb{R}_+^M$ with $|u|_{\infty} \le t_0(r, K)$ the following holds:

$$ \mathcal{L}(u) \le \mathcal{L}(|u|_{\infty} \mathbf{1}) \le |u|_{\infty} \left(1 + \frac{2K}{1-r}\right) \mathbf{1} \le \log\left(\frac{1+r}{2r}\right) \mathbf{1}, \quad (3.7) $$
$$ \mathcal{L}(u) \preceq \left( \mathrm{Id} - \frac{1+r}{2r} \mathbf{H} \right)^{-1} u. \qquad (3.8) $$

**Remark 3.2.** The whole information on the exponential moments is contained in the fixed point Equation (3.3).

**Remark 3.3.** As explained in Theorem 5.6 in Section 5, if $|\mathcal{L}(u)|_\infty < \infty$ and $\mathrm{SpR}(\mathbf{H}) < 1$, then the condition $\mathrm{SpR}(\mathbf{H} \mathrm{diag}(e^{\mathcal{L}(u)})) < 1$ holds except if $u$ is a boundary point, meaning that for any $t > 1$ we have $|\mathcal{L}(tu)|_\infty = \infty$. Of course even if the condition $\mathrm{SpR}(\mathbf{H} \mathrm{diag}(e^{\mathcal{L}(u)})) < 1$ does not hold we still have

$$ \mathcal{L}(u) \preceq \sum_{n \ge 0} (\mathbf{H} \mathrm{diag}(e^{\mathcal{L}(u)}))^{n} u, $$

which is actually the result proved in the proof. In (3.8), since $\mathbf{H} \in \mathbf{Ge}(r, K)$, we have $\mathrm{SpR}(\mathbf{H}) \le r$ and thus for $|u|_\infty \le t_0(r, K)$, it follows that $\mathrm{SpR}(\frac{1+r}{2r}\mathbf{H}) \le \frac{1+r}{2} < 1$, which allows to give sense of the inverse.

**Remark 3.4.** If one choose $u = te_{m_0}$ with $t \le t_0(r, K)$, then (3.8) says that $\mathcal{L}(te_{m_0})$ is smaller than the column $m_0$ of $t(\mathrm{Id} - \frac{1+r}{2r}\mathbf{H})^{-1}$. Thus the exponential moment of the number of individuals of type $m_0$ of a Poissonian Galton Watson process with root of type $m$ is bounded by the exponential of the entry $(m, m_0)$ of $t(\mathrm{Id} - \frac{1+r}{2r}\mathbf{H})^{-1}$. In comparison, the global estimate, by taking $u = t\mathbf{1}$ and a root of type $m \in \mathcal{M}$, gives the sum over the row $m$ of $t(\mathrm{Id} - \frac{1+r}{2r}\mathbf{H})^{-1}$, which is, roughly speaking, $M$ times bigger than the entry $(m, m_0)$.

One may ask the following question, is the condition $\mathrm{SpR}(\mathbf{H}) < 1$ necessary to have the existence of non zero $u \in \mathbb{R}_+^M$ such that $|\mathcal{L}(u)|_\infty < \infty$? We refer the reader to Section 5 for more details on this but the short answer is no since one can construct trivial example of this type. Let

$$ \mathbf{H} = \begin{pmatrix} \alpha & 0 \\ 0 & 0 \end{pmatrix} $$

with $\alpha \ge 1$ and consider $u = e_2$ then clearly, $\mathcal{L}(e_2) = e_2$ which is finite and $\mathrm{SpR}(\mathbf{H}) = \alpha \ge 1$. Also in this case, $\mathcal{L}(e_1) = (\infty, 0)$ which is infinite only on the first coordinate. We need an assumption on $\mathbf{H}$ such as *irreducibility*. We recall that a square matrix $A = (A_{ij})_{i,j \in I}$ with non negative coefficients is irreducible if the directed graph with vertices $I$ and containing the edges $(i \to j)$ such that $A_{ij} > 0$ is strongly connected, which is equivalent to the following property,

$$ \exists N \ge 1, (\mathrm{Id} + A)^N \succ 0. $$

**Lemma 3.5** stated below gives a lower bound on $\mathcal{L}$ and first results on domains of finiteness and infiniteness of the function $\mathcal{L}$. It is proved in Section 6.2.

**Lemma 3.5.** Let $u \in \mathbb{R}_+^M$, and $\mathcal{T}^m$ a Poisson(\(\mathbf{H}\)) Galton Watson process. Then we have,

$$ \left( \mathrm{E} \left[ u \cdot \mathrm{Card}_{\mathcal{M}}(\mathcal{T}^m) \right] \right)_{m \in \mathcal{M}} = \sum_{n \ge 0} \mathbf{H}^n u \preceq \mathcal{L}(u) - \mathbf{1}. \qquad (3.9) $$

In particular the following holds.

(1) If we suppose that $\mathbf{H}$ is irreducible and that $\mathrm{SpR}(\mathbf{H}) \ge 1$. Then for any $u \in \mathbb{R}_+^M$ and $u \ne 0$ we have

$$ \forall m \in \mathcal{M}, \mathcal{L}(u)_m = \infty. $$
(2) If there exists $u > 0$ such that $|\mathcal{L}(u)|_\infty < \infty$ then we have $\text{SpR}(\mathbf{H}) < 1$.

Equality (3.9) is well-known and not specific to the Poissonian framework. We give its proof only for completeness. Lower bound (3.9) on $\mathcal{L}(u)$ is just convexity of the exponential. Irreducibility conditions are classical when one is interested in probabilities of extinction, see for example [7], and are thus linked to finiteness condition of moments and exponential moments of Poissonian Galton Watson processes. For more details we refer the reader to Section 5.

In the one dimensional case we can find explicitly the greatest $u \in \mathbb{R}_+$ such that the exponential moment is finite. This result is stated below in Theorem 3.6 and proved in Section 6.3.

**Theorem 3.6.** Let $\mathcal{T}$ a single type Poisson(α) Galton Watson process with $0 < \alpha < 1$. Let

$$u_c = \log (1/\alpha) - (1 - \alpha). \qquad (3.10)$$

Then, for $u \ge 0$ we have

$$\mathbb{E} [e^{u \text{Card}(\mathcal{T})}] < \infty \iff u \le u_c, \qquad (3.11)$$

and we have

$$\mathbb{E} [e^{u_c \text{Card}(\mathcal{T})}] = \frac{1}{\alpha}. \qquad (3.12)$$

Finally, for any $x \ge 0$ such that $u := x - \alpha(e^x - 1) \ge 0$ the following holds,

$$x = \mathcal{L}(u) \iff \alpha e^x \le 1. \qquad (3.13)$$

This result is of interest since it is not clear a priori that the exponential moment at $u_c = \sup\{u \ge 0 \mid \mathcal{L}(u) < \infty\}$ is finite. Also, by (3.13), for any $0 \le x \le 1/\alpha$ we have $\mathcal{L}(x - \alpha(e^x - 1)) = x$, which, up to an inverse problem, completely solves the question of the exponential moments in the one dimensional case. In Section 5 we derive the multitype extension of Theorem 3.6 and perform a complete study of the topological properties of the set of the $u$'s where $\mathcal{L}(u)$ is finite.

## 3.2. Application to Hawkes processes.
Consider $\mu := (\mu_m)_{m \in \mathcal{M}} \in \mathbb{R}_+^M$ and suppose that the functions $h_{ij}$ only depend on $t-s$ instead of $(s,t)$, i.e. that we have $h_{ij}(s,t) = h_{ij}(t-s)$ for any $i,j \in \mathcal{M}$.

Let $(\Omega, \mathcal{F}, \mathbb{P})$ a probabilistic space and a filtration $(\mathcal{F}_t)_{t \in \mathbb{R}}$. A Hawkes process with parameters $\mathcal{M}, \mu, h$ is a multivariate point process $\mathcal{N} := (N^m)_{m \in \mathcal{M}}$ such that

* $\mathcal{N}$ is adapted to $(\mathcal{F}_t)_{t \in \mathbb{R}}$,

* almost surely, for $m \ne m'$ in $\mathcal{M}$, $N^m$ has no common points with $N^{m'}$,

* $\mathcal{N}$ has a predictable intensity $\lambda := (\lambda^m)_{m \in \mathcal{M}}$ given by

$$\lambda_t^m = \mu_m + \sum_{m' \in \mathcal{M}} \int_{-\infty}^{t^-} h_{m',m}(t-s) dN_s^{m'}, \quad m \in \mathcal{M}, t \in \mathbb{R}. \qquad (3.14)$$

In [4] it is proved that a process defined as above exists and is stationary under the condition that the spectral radius of the matrix $\mathbf{H} = (\|h_{m,m'}\|_1)_{m,m' \in \mathcal{M}}$ is strictly smaller than 1, ie $\text{SpR}(\mathbf{H}) < 1$.

For $A \subset \mathbb{R}$ we denote $\mathcal{N}(A) = (N^m(A))_{m \in \mathcal{M}}$ the number of point in $A$ of each type. Theorem 3.1 combined with the method developed in [11] leads to Theorem 3.7 stated below and proved in Section 6.4.
**Theorem 3.7.** Let $u \in \mathbb{R}_+^M$ and $\mathcal{L}(u)$ from Theorem 3.1 with $\mathbf{H} = (\|h_{m,m'}\|_1)_{m,m' \in M}$. The following holds,

$$ \forall L \ge 0, \quad \mathbb{E} [e^{u \cdot \mathcal{N}(|0,L|)}] \le \exp \left( L \boldsymbol{\mu} \cdot \left( e^{\mathcal{L}(u)} - \mathbf{1} \right) \right). \quad (3.15) $$

More precisely, suppose that $\mathbf{H} \in \mathbf{Ge}(r, K)$ with $r < 1$, then for all $u \in \mathbb{R}_+^M$ such that $|u|_\infty \le t_0(r, K)$ we have

$$ \forall L \ge 0, \quad \mathbb{E} \left[ e^{u \cdot \mathcal{N}(|0,L|)} \right] \le \exp \left[ L \boldsymbol{\mu} \cdot \left( e^{\left( \text{Id} - \frac{1+r}{2r} \mathbf{H} \right)^{-1} u} - \mathbf{1} \right) \right] \quad (3.16) $$

where $t_0(r, K)$ is defined in Theorem 3.1.

**Remark 3.8.** Theorem 1.4 of [11] is a particular case of Theorem 3.7 corresponding to taking $u = t\mathbf{1}$ with $0 \le t \le t_0(r, K)$. The case $u = t\mathbf{1}$ results in an exponential estimate for all the points regardless of their types. Theorem 3.7 allows to take $u = t\mathbf{e}_m$ for example leading to a better bound if one is interested only in the points of type $m$.

**Remark 3.9.** One can always use the bound $e^x - \mathbf{1} \succeq \text{diag}(e^x)x$ for any $x \in \mathbb{R}_+^M$ to obtain the bound

$$ \forall L \ge 0, \quad \mathbb{E} \left[ e^{u \cdot \mathcal{N}(|0,L|)} \right] \le \exp \left( L \boldsymbol{\mu} \cdot \text{diag} \left( e^{\mathcal{L}(u)} \mathcal{L}(u) \right) \right) $$

to get rid of one of the nested exponentiation.

**3.3. Tail estimates for Galton Watson processes.** In Section 3.1 we derived exponential moments for individuals of specific types for $\mathbf{M}$-type Galton Watson processes. We are now interested in the decay rate of the tail of a Galton Watson process. More precisely, we are interested in individuals of a Galton Watson process whose generations are at least $n$ with $n \to \infty$. Theorem 3.10 states exponential estimates for the tails of Galton Watson processes in a framework that keeps the information on the types. Theorem 3.10 is proved in Section 6.5.

**Theorem 3.10.** Let $\mathcal{T}^m$ a Poisson($\mathbf{H}$) Galton Watson processes with root of type $m \in \mathbf{M}$. Let $u \in \mathbb{R}_+^M$. Define the sequence $(\mathcal{R}_n(u))_{n \in \mathbb{N}} \in [0, \infty]^\mathbb{N}$ by

$$ \forall n \in \mathbb{N}, \forall m \in \mathbf{M}, \quad \exp \left( e_m \cdot \mathcal{R}_n(u) \right) = \mathbb{E} \left[ e^{u \cdot \text{Card}_M \left( \mathcal{T}_{\ge n}^m \right)} \right]. \quad (3.17) $$

Then the sequence $(\mathcal{R}_n(u))_{n \in \mathbb{N}}$ satisfies the following fixed point equation,

$$ \mathcal{R}_0(u) = \mathcal{L}(u) \quad \text{and} \quad \mathcal{R}_{n+1}(u) = \mathbf{H} \left( e^{\mathcal{R}_n(u)} - \mathbf{1} \right), \quad n \ge 0, \quad (3.18) $$

where $\mathcal{L}(u)$ is defined in Theorem 3.1. For any $n \ge 0$ the following holds,

$$ \mathcal{R}_{n+1}(u) \preceq \mathcal{R}_n(u) \preceq \mathcal{R}_0(u) = \mathcal{L}(u). \quad (3.19) $$

If we suppose that $\text{SpR}(\mathbf{H}) < 1$ and $|\mathcal{L}(u)|_\infty < \infty$, then there exists a constant $C$ depending on both $u$ and $\mathbf{H}$ such that for all $n \in \mathbb{N}$,

$$ \mathcal{R}_n(u) \preceq C H^n \mathcal{L}(u). \quad (3.20) $$

More precisely, if we suppose that $\mathbf{H} \in \mathbf{Ge}(r, K)$ with $r < 1$, then for all $u \in \mathbb{R}_+^M$ with $|u|_\infty \le t_0(r, K)$ and for all $n \ge 0$, we have

$$ \mathcal{R}_n(u) \preceq \left( \frac{1+r}{2r} \right)^{1+\frac{2K}{1-r}} \mathbf{H}^n \mathcal{L}(u), \quad (3.21) $$

where $t_0(r, K)$ is defined in Theorem 3.1.

**Remark 3.11.** Once again, a fix point equation, on the sequence $(\mathcal{R}_n(u))_{n \in \mathbb{N}}$, contains the whole information about exponential moments of the tails of Galton Watson processes.
Remark 3.12. If $\mathbf{H} \in \mathbf{Ge}(r, K)$ with $r < 1$ and if $u \in \mathbb{R}_+^M$ with $|u|_\infty \le t_0(r, K)$, since $\|\mathbf{H}^n\|_\infty \le K r^n$ with $r < 1$ we have that $|\mathcal{R}_n(u)|_\infty \le K'r^n|u|_\infty$ with $K'$ a constant. Therefore $\mathcal{R}_n(u)$ decays exponentially fast.

Theorem 3.10 is a straightforward application of Theorem 4.5, the extension of Theorem 4.1, about tail estimates for clusters presented in the following section. The idea is to consider a cluster where the birth date of an individual is exactly its generation. One could prove Theorem 3.10 directly but the proof would be a specific case of the proof of Theorem 4.5 and thus we only present the proof in the general case of Theorem 4.5.

## 4. Tail estimates for inhomogeneous Poisson clusters

4.1. Main result for inhomogeneous Poisson clusters. In Section 3 we derived exponential estimates for $M$-type Galton Watson processes, how it applies to Hawkes processes and exponential estimates for the tail in terms of generations of a Galton Watson process. In this section we are studying inhomogeneous Poisson clusters generated by bivariate functions $\mathbf{h} = (h_{ij})_{i,j \in M}$ and we denote $\mathbf{H} = \|\mathbf{h}\|_{L^\infty L^1}$. A cluster consists in an $M$-type random tree with birth dates attached to each individuals. If one omits the birth dates it is clear that the resulting $M$-type random tree is stochastically dominated by a $\mathcal{P}$ois($\mathbf{H}$) Galton Watson process, which means that there exists a coupling for which the second one contains the first one. Thus the results of Section 3 immediately transfer to clusters. The main question on clusters lies in the repartition of the birth dates. In this section we state exponential estimates for the temporal tail of clusters: the number of individuals with birth dates in intervals of the form $[t, \infty)$ with $t \to \infty$. Technical results on bivariate convolution are necessary for the proof of the main result of this section, Theorem 4.1. These results are stated in Appendix A.

We consider inhomogeneous and $M$-type Poisson clusters generated by the functions $\mathbf{h}(s, t)$ where $\mathbf{h}(s, t)$ are non negative. We also denote

$$ \mathbf{H} = \|\mathbf{h}\|_{L^\infty L^1}. $$

Theorem 4.1 stated below gives exponential estimates on the tail of inhomogeneous Poissonian clusters. We also quantify how fast this quantity decays given the decay of the interaction functions $\mathbf{h}$. Theorem 4.1 is proved in Section 6.6.

**Theorem 4.1.** Let $G_s^m$ a $\mathcal{P}$ois($\mathbf{h}$) cluster born at time $s \in \mathbb{R}$ with root of type $m \in M$. Let $u \in \mathbb{R}_+^M$ and for $t \in \mathbb{R}$ define

$$ f_{u,m}(s,t) = \log \left( \mathbb{E} \left[ e^{u \cdot \text{Card}_M \left( G_s^m \cap [t, \infty) \right)} \right] \right). $$

Then, $\mathbf{f}_u = (f_{u,m})_{m \in M}$ satisfies the following fixed point equation,

$$ \mathbf{f}_u(s,t) = u \mathbf{1}_{t \le s} + \left[ \mathbf{h} \star (e \mathbf{f}_u - \mathbf{1}) \right](s,t), \quad s, t \in \mathbb{R}. \quad (4.1) $$

Let $\mathcal{L}(u)$ from Theorem 3.1 with $\mathbf{H} = \|\mathbf{h}\|_{L^\infty L^1}$. Then for any $s, t \in \mathbb{R}$ we have $\mathbf{f}_u(s,t) \le \mathcal{L}(u)$ and the following bound holds,

$$ \mathbf{f}_u(s,t) \le u \mathbf{1}_{t \le s} + R \Psi_u(s,t) \times u, \quad (4.2) $$

where

$$ \Psi_u = \sum_{n=1}^{\infty} (\mathbf{h} \operatorname{diag}(e^{\mathcal{L}(u)}))^{*n}. $$

In particular the following holds.
(1) Suppose that $\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)}) \in \mathbf{Ge}(\alpha, K)$ for some $\alpha < 1$ and that

$$R_h^\infty(d) \preceq Ae^{-cd}\mathbf{H}, \quad \forall d \geq 0$$

for some $A$ and $c > 0$. Then for all $0 < \varepsilon < \frac{c(1-\alpha)}{(1-\alpha)+A\alpha}$ we have

$$|\mathbf{f}_u(s, s+d)|_\infty \le B_\varepsilon e^{-\varepsilon d}|u|_\infty, \quad \forall s \in \mathbb{R}, \forall d > 0,$$

with $B_\varepsilon = \frac{K\alpha(1+\frac{A\varepsilon}{c-\varepsilon})}{1-\alpha(1+\frac{A\varepsilon}{c-\varepsilon})}$.

(2) Suppose that $\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)}) \in \mathbf{Ge}(\alpha, K)$ for some $\alpha < 1$ and that

$$R_h^\infty(d) \preceq \frac{A}{(1+d)^\gamma}\mathbf{H}, \quad \forall d \geq 0$$

for some $A$ and $\gamma > 0$. Then for any $0 < \delta < 1$ we have

$$|\mathbf{f}_u(s, s+d)|_\infty \le \frac{B_\delta|u|_\infty}{(1+d)^{\gamma(1-\delta)}}, \quad \forall s \in \mathbb{R}, \forall d \ge 1,$$

with $B_\delta = \frac{\alpha K(K+1)}{(1-\alpha)^2} \left[ A \left( 1 + \frac{\gamma}{\delta \log(1/\alpha)} \right)^\gamma + 2^\gamma \right]$.

(3) Suppose that $\operatorname{SpR}(\mathbf{H}) < 1$, that $|\mathcal{L}(u)|_\infty < \infty$ and that $R_h^\infty(A) = 0$ or equivalently that $\mathbf{h}$ is essentially supported on $\{(s,t) \in \mathbb{R}^2 \mid t \le s + A\}$ for some $A > 0$. Then there exists a constant $C$ depending on both $u, \mathbf{H}$ and $A$ such that

$$\mathbf{f}_u(s, s+d) \preceq C\mathbf{H}^{[d/A]}\mathcal{L}(u), \quad \forall d \ge 0.$$

If $\mathbf{H} \in \mathbf{Ge}(r, K)$ with $r < 1$ and $|u|_\infty \le t_0(r, K)$ from Theorem 3.1, then in points (1) and (2) one can take $\alpha = \frac{1+r}{2}$ and $K = K$. In point (3) one can take $C = (\frac{1+r}{2r})^{1+2K/(1-r)}$.

**Remark 4.2.** Point (1) of Theorem 4.1 proves that if $\mathbf{h}$ decays exponentially fast then it is also the case for the tails of the clusters but with a smaller rate. Similarly point (2) proves that if $\mathbf{h}$ decays according to a power law then the tails decay with a power as close as we want of the one of $\mathbf{h}$. Finally, point (3) proves that if $\mathbf{h}$ is supported in $\{(s,t) \in \mathbb{R}^2 \mid t \le s + A\}$ then $\mathbf{f}_u$ also has exponential decay, with rate given by $\mathbf{H}$.

**Remark 4.3.** If $R_h^\infty(A) = 0$ for some $A > 0$, then from direct support considerations and Equation (4.2), one can derive that

$$\mathbf{f}_u(s, s+d) \preceq \sum_{n \ge |d/A|} (\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)}))^{n} u, \quad \forall d \ge 0.$$

Since $\mathcal{L}(u) \preceq \sum_{n \ge 0} (\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)}))^{n} u$ it is clear that the result of point (3) is in some sense better (up to the constant $C$) since we have

$$\mathbf{H}^{[d/A]} \preceq (\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)}))^{[d/A]}.$$

If $\mathcal{L}(u) > 0$ then it is clear that for $d$ large enough we have

$$C\mathbf{H}^{[d/A]} \preceq (\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)}))^{[d/A]}.$$

**Remark 4.4.** The condition $\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)}) \in \mathbf{Ge}(\alpha, K)$ with $\alpha < 1$ is equivalent to the condition $\operatorname{SpR}(\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)})) < 1$ and thus, as explained in Remark 3.3, Theorem 5.6 proves that if $\operatorname{SpR}(\mathbf{H}) < 1$ and $\mathcal{L}(u)$ is finite then $\operatorname{SpR}(\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)})) < 1$ holds except if $u$ is a boundary point.
4.2. Extension of clusters and application to generation's tail of Galton Watson. One can extend the definitions of clusters by allowing more general distributions than distributions with densities with respect to the Lebesgue measure. Indeed, for all $s \in \mathbb{R}$ the functions $h_{ij}(s, \cdot)$ are proportional to the densities of the law of the birth date of individual of type $j$ with a parent of type $i$ born at time $s$. Thus the natural extension is to consider $h_{ij}(s, \cdot)$ as given non negative measures on $\mathbb{R}$ for all $s \in \mathbb{R}$. To avoid any confusion we denote $\varphi_{ij}$ these objects and $\varphi = (\varphi_{ij})_{i,j \in M}$. A $\mathcal{P}$oïs($\varphi$) cluster is then defined by the following:

* Independently for any $j$, an individual $a$ of type $i$ and born at time $s$ has $\mathcal{P}$oïs($\varphi_{ij}(s, \mathbb{R})$) children of type $j$.

* Independently, for any child $b$ of $a$ of type $j$, we have

$$ \mathrm{bd}(b) \sim \frac{\varphi_{ij}(s, \cdot)}{\varphi_{ij}(s, \mathbb{R})}. $$

To avoid measurability complications we only consider this extension in the homogeneous case where for any borelian set $A$, for any $i, j \in M$, we have that $\varphi_{ij}(s, A)$ depends only on $A - s$. With a slight abuse of notation, we still call $\varphi_{ij}$ the measure that satisfies

$$ \varphi_{ij}(s, A) = \varphi_{ij}(A - s). $$

The matrix $H$ is similarly defined by $H_{ij} = \varphi_{ij}(\mathbb{R})$. Again, if one omits the birth dates of such a cluster, then one recovers a $\mathcal{P}$oïs($H$) Galton Watson process. In the inhomogeneous case one has to tackle issues such as measurability of $s \in \mathbb{R} \mapsto \varphi_{ij}(s, \cdot) \in \mathcal{M}(\mathbb{R})$ with $\mathcal{M}(\mathbb{R})$ the set of finite measures on $\mathbb{R}$.

Since we are in the homogeneous case, without loss of generality we can suppose that the root of the cluster is born at $s = 0$ because the law of a cluster born at time $s$ is the law of a cluster born at $s = 0$ with all the birth dates translated by $s$.

Let us recall the usual convolution product between two measures $\mu$ and $\nu$ on $\mathbb{R}$: for all $A$ measurable set,

$$ \mu \star \nu(A) = \int_{\mathbb{R}^2} 1_{x+y \in A} d\mu(x)d\nu(y). $$

Similarly, one can define the convolution between a measure $\mu$ and a real measurable function $f$ as follows,

$$ \mu \star f(t) = \int_{\mathbb{R}} d\mu(x)f(t-x), \quad \forall t \in \mathbb{R}. $$

With these definitions, Theorem 4.1 holds in the exact same formulation with the exact same proof, which we do not write again.

**Theorem 4.5.** Let $G^m$ a homogeneous $\mathcal{P}$oïs($\varphi$) cluster born at time 0 with root of type $m \in M$. Let $u \in \mathbb{R}_+^M$ and for $t \in \mathbb{R}$ define

$$ \mathbf{f}_{u,m}(t) = \log \left( \mathbb{E} \left[ e^{u \cdot \mathrm{Card}_M (G^m \cap [t, \infty))} \right] \right). $$

Then, $\mathbf{f}_u = (f_{u,m})_{m \in M}$ satisfies the following fixed point equation,

$$ \mathbf{f}_u(t) = u \mathbf{1}_{t \le 0} + \left[ \varphi \star (e^{\mathbf{f}_u} - \mathbf{1}) \right](t), \quad t \in \mathbb{R}. \qquad (4.3) $$

Let $\mathcal{L}(u)$ from Theorem 3.1 with $\mathbf{H} = \varphi(\mathbb{R})$ and suppose that $|\mathcal{L}(u)|_\infty < \infty$. Then for any $t \in \mathbb{R}$ the following bound holds,

$$ \mathbf{f}_u(t) \le u \mathbf{1}_{t \le 0} + \mathbf{\Psi}_u([t, \infty)) \times u, \qquad (4.4) $$
where the measure $\Psi_u$ is defined by

$$ \Psi_u = \sum_{n=1}^{\infty} (\varphi \operatorname{diag}(e^{\mathcal{L}(u)}))^{*n}. $$

Points (1), (2) of Theorem 4.1 also hold up to replacing $R_h^\infty(d)$ by $\varphi([d, \infty))$ for any $d \in \mathbb{R}$ and for point (3), one should ask $\varphi$ to be supported in $(-\infty, A]$.

The main application of this extension to general distributions is that we can take some deterministic distributions for the birth dates. For example, let $\varphi = \delta_1 H$ with $\delta_1$ the Dirac mass at 1, which means that

$$ \varphi_{ij} = H_{ij}\delta_1, \quad \forall i, j \in \mathcal{M}. $$

Then, for clusters born at time $s = 0$, the birth date of an individual of the $n^{th}$ generation is exactly $n$.

Thus in this case, it is clear that for any $n \ge 0$, we have $f_u(n) = \mathcal{R}_n(u)$ from Theorem 3.10. Thus Theorem 3.10 is a direct consequence of the extended version of Theorem 4.1: the fixed point equation is (4.3) and the bound is point (3) applied with $A = 1$.

## 5. Exact study of type specific exponential moments for Poissonian Galton Watson processes.

This section aims to understand the analytic properties of the exponential moments of a $\mathcal{Poi}_s(H)$ Galton Watson tree with types in $M$ and gives a multitype extension of Theorem 3.6. In the one dimensional case there is only one possible direction for the linear combination $u$, and thus the complexity of the structure of the exponential moments is relatively low. In the multitype case the situation is a little less clear-cut but we still manage to obtain interesting results, stated in Theorem 5.6.

We recall that $H$ is a $M \times M$ matrix with non negative coefficients and for any $u \in \mathbb{R}_+^M$ the vector $\mathcal{L}(u)$ is defined by

$$ \exp (e_m \cdot \mathcal{L}(u)) = \mathbb{E} \left[ e^{u \cdot \mathrm{Card}_M(\mathcal{T}^m)} \right] $$

where $\mathcal{T}^m$ is a $\mathcal{Poi}_s(H)$ Galton Watson tree with root of type $m \in M$, see Theorem 3.1.

The following definition introduces the key objects for the multitype study.

**Definition 5.1.** Let $E \subset \mathbb{R}_+^M$ defined by

$$ E = \{u \in \mathbb{R}_+^M \mid |\mathcal{L}(u)|_{\infty} < \infty\}. \qquad (5.1) $$

Thus $u \in \mathbb{R}_+^M$ belongs to $E$ if and only if there exists $x \in \mathbb{R}_+^M$ such that $x = u + H(e^x - 1)$. Let also

* Let $\delta E := \{u \in E \mid \forall t > 1, tu \notin E\}$.

* Let $\tilde{E} := E \setminus \delta E \subset \mathbb{R}_+^M$.

* Let $\tilde{E}^* = \{u \in \tilde{E} \mid u > 0\} \subset (0, \infty)^M$.

For any set $F \subset \mathbb{R}^M$ define the lower cone of $F$ for $\le$ by

$$ \mathrm{LC}_{\leq}(F) = \{u \in \mathbb{R}^M \mid \exists v \in F, u \leq v\}. $$
Remark 5.2. These notations are inspired from topology and we will show that these are, in some sense, coherent with the classical ones in the case SpR($\mathbf{H}$) < 1. Figure 5.1 gives a visual representation of $\delta E$, $\mathring{E}$ and $\mathring{E}^*$.

As we saw in Lemma 3.5, irreducibility may play an important role. The following definition introduces key objects to understand the structure of the interactions when $\mathbf{H}$ is not irreducible.

**Definition 5.3.** For $m \in \mathbf{M}$ define $\mathcal{C}(m) \subset \mathbf{M}$ the connected component of $m$ by

$$ \mathcal{C}(m) = \{m' \in \mathbf{M} \mid \exists N \ge 0, (\mathbf{H}^N)_{m,m'} > 0\}. \quad (5.2) $$

In particular $m \in \mathcal{C}(m)$. For $m' \neq m$, then $m'$ belongs to $\mathcal{C}(m)$ if and only if it is possible for an individual of type $m$ to have an offspring of type $m'$, possibly after more than one generation. For any $m \in \mathbf{M}$ we define

$$ \mathbf{H}_{\|\mathcal{C}(m)}} = (H_{ij})_{i,j \in \mathcal{C}(m)} \in \mathbb{R}^{\mathcal{C}(m)^2}. \quad (5.3) $$

Remark 5.4. For any $m \in \mathbf{M}$ and any vector $x \in \mathbb{R}^M$ with support in $\mathcal{C}(m)$, then $\mathbf{H}x$ has support in $\mathcal{C}(m)$ and if we denote $x_{|\mathcal{C}(m)}} = (x_i)_{i \in \mathcal{C}(m)}}$ we have the following identity in $\mathbb{R}^{\mathcal{C}(m)}}$,

$$ \mathbf{H}_{\|\mathcal{C}(m)}} x_{|\mathcal{C}(m)}} = (\mathbf{H}x)_{|\mathcal{C}(m)}}. $$

The following also holds: for any $m, m' \in \mathbf{M}$, if $m' \in \mathcal{C}(m)$ then $\mathcal{C}(m') \subset \mathcal{C}(m)$. Also, if $\mathbf{H}$ is irreducible then $\mathcal{C}(m) = \mathbf{M}$ for any $m \in \mathbf{M}$.

The following Lemma is clear given Definition 5.3.

**Lemma 5.5.** Let $m \in \mathbf{M}$, and $u \in \mathbb{R}_+^M$ with support in $\mathcal{C}(m)$. Then $\mathcal{L}(u)$ also has support in $\mathcal{C}(m)$ and if we denote $\mathcal{L}(\mathbf{H}_{\|\mathcal{C}(m)}}, \mathcal{C}(m))$ the equivalent of $\mathcal{L}$ but with $(\mathbf{H}_{\|\mathcal{C}(m)}}, \mathcal{C}(m))$ instead of $(\mathbf{H}, \mathbf{M})$ we have the following equality,

$$ (\mathcal{L}(u))_{|\mathcal{C}(m)}} = \mathcal{L}(\mathbf{H}_{\|\mathcal{C}(m)}}, \mathcal{C}(m)) (u_{|\mathcal{C}(m)})). $$

There is nothing to prove in Lemma 5.5, since for any $m' \in \mathcal{C}(m)$ a $\mathcal{P}$oïs($\mathbf{H}$) Galton Watson process with root of type $m'$ has only individuals with types in $\mathcal{C}(m)$. Therefore for any $m' \in \mathbf{M}$, a $\mathcal{P}$oïs($\mathbf{H}_{\|\mathcal{C}(m)}$) Galton Watson process with root of type $m'$ has the same distribution as a $\mathcal{P}$oïs($\mathbf{H}$) Galton Watson process with root of type $m'$.

Let us now state the main result of this section, Theorem 5.6, proved in Section 6.7.

**Theorem 5.6.** The following holds.

(1) $E$ is convex, closed and if $0 \preceq u \in E$ then for any $0 \preceq v \preceq u$ we have $v \in E$, which is equivalent to $E = \text{LC}_\preceq(E) \cap \mathbb{R}_+^M$.

(2) We have $\mathring{E}^* \neq \emptyset \iff \text{SpR}(\mathbf{H}) < 1$. Thus if $\text{SpR}(\mathbf{H}) < 1$ we have that $\mathring{E}$ is the interior of $E$ for the induced topology of $\mathbb{R}^M$ on $\mathbb{R}_+^M$ and $\delta E$ is the boundary of $E$ for the same topology.

(3) We have $\text{SpR}(\mathbf{H}_{\|\mathcal{C}(m)}) < 1 \iff \exists \varepsilon > 0, \varepsilon e_m \in E$ and thus the following holds, $E = \{0\} \iff \forall m \in \mathbf{M}$, $\text{SpR}(\mathbf{H}_{\|\mathcal{C}(m)}) \ge 1$.

(4) The application $u \in E \mapsto \mathcal{L}(u)$ is bijective onto its image, continuous on $\mathring{E}$ and any $\text{LC}_\preceq(\{u\})$ for $u \in E$. If $\text{SpR}(\mathbf{H}) < 1$ the application $u \in \mathring{E} \mapsto \mathcal{L}(u)$ is infinitely differentiable.
(5) For any $u \in \mathbb{R}_+^M$, there exists at most one solution of the equation $x = u + \mathbf{H}(e^x - 1)$ with $x \in \mathbb{R}_+^M$ and $\text{SpR}(\mathbf{H} \text{diag}(e^x)) \le 1$. Thus if such solution $x$ exists, we must have $x = \mathcal{S}(u)$.

(6) If $\text{SpR}(\mathbf{H}) < 1$ we have

$$ \begin{aligned} \dot{E} &= \{u \in \mathbb{R}_+^M \mid \exists y \in \mathbb{R}_+^M, u = y - \mathbf{H}(e^y - 1), \text{SpR}(\mathbf{H} \text{diag}(e^y)) < 1\} \\ &= \{u \in E \mid \text{SpR}(\mathbf{H} \text{diag}(e^{\mathcal{S}(u)})) < 1\}. \end{aligned} $$

(7) If $\text{SpR}(\mathbf{H}) < 1$ we have

$$ \begin{aligned} \delta E &= \{u \in \mathbb{R}_+^M \mid \exists y \in \mathbb{R}_+^M, u = y - \mathbf{H}(e^y - 1), \text{SpR}(\mathbf{H} \text{diag}(e^y)) = 1\} \\ &= \{u \in E \mid \text{SpR}(\mathbf{H} \text{diag}(e^{\mathcal{S}(u)})) = 1\}. \end{aligned} $$

*Remark 5.7.* Points (6) and (7) allow to construct exact solutions for the exponential moments, and even maximal ones with the condition $\text{SpR}(\mathbf{H} \text{diag}(e^y)) = 1$, by guessing values $y$ for $\mathcal{S}(u)$. However, one does not control the value and the direction obtained for the associated $u$.

Figure 5.1 gives a schematic representation of the topological properties of the set $E$ when $\text{SpR}(\mathbf{H}) < 1$.

<figure><img src="image_2.png" /><figcaption>FIGURE 5.1. Illustration of the topological properties of $E$ when $\text{SpR}(\mathbf{H}) < 1$. By Remark 5.8 we can always assume it is the case up to removing axis where $\mathcal{S}$ is not finite. The set $\delta E$ is represented in red while $\dot{E}$ consists in the union of the blue and light blue parts. The light blue part is $\dot{E}^*$. The point $u_c$ is a critical value of $u$ since it is the greatest $u$ in the direction $\frac{u_c}{\|u_c\|}$ for which $\mathcal{S}$ is finite, which is the definition of being part of $\delta E$.</figcaption></figure>

*Remark 5.8.* Most of the above results are stated under the assumption $\text{SpR}(\mathbf{H}) < 1$. If it is not the case then by (3) there are two possibilities. Either $E = \{0\}$, either there exists $m \in \mathcal{M}$ such that $\text{SpR}(\mathbf{H}_{\|\mathcal{C}(m)\}) < 1$. Suppose that $E \neq \{0\}$. Then consider the set of types $\tilde{\mathcal{M}} \subset \mathcal{M}$ defined by

$$ \tilde{\mathcal{M}} = \{m \in \mathcal{M} \mid \text{SpR}(\mathbf{H}_{\|\mathcal{C}(m)\}) < 1\}. $$

From remark 5.4 it is clear that

$$ \tilde{\mathcal{M}} = \bigcup_{m \in \tilde{\mathcal{M}}} \mathcal{C}(m), $$
and thus for any $x$ with support in $\tilde{M}$, then $Hx$ has also support in $\tilde{M}$ and we have

$$ (Hx)|_{\tilde{M}} = H_{\|\tilde{M}}x|_{\tilde{M}}. $$

Again by (3) it is clear that $E$ lies entirely in the set

$$ \{u \in \mathbb{R}_+^M \mid \forall m \in M \setminus \tilde{M}, u_m = 0\}. $$

Thus one can perform the study of $E$ as a subset of $\mathbb{R}_+^{\tilde{M}}$ with $(\tilde{M}, H_{\|\tilde{M}})$ instead of $(M, H)$. By definition of $\tilde{M}$ it is clear that there exists $\varepsilon > 0$ such that for any $m \in \tilde{M}$ we have $\varepsilon e_m \in E$. Thus by convexity, up to reducing $\varepsilon$, we have $(\varepsilon)_{m \in \tilde{M}} \in E$ and thus by point (2) we have $\mathrm{SpR}(H_{\|\tilde{M}}) < 1$ and all the results applies for $(\tilde{M}, H_{\|\tilde{M}})$.

## 6. Proofs

Before diving in the proof let us recall the classical Perron-Frobenius Theorems.

**Perron-Frobenius Theorem:** Let $A$ a square matrix of size $N$, with non negative coefficients and irreducible. Let $\rho = \mathrm{SpR}(A)$. Then there exists $\kappa \in (0, \infty)^N$ such that

$$ A\kappa = \rho\kappa. $$

We can avoid the irreducible condition up to a weaker conclusion, which is the theorem stated below.

**Weak Perron-Frobenius Theorem:** Let $A$ a square matrix of size $N$, with non negative coefficients. Let $\rho = \mathrm{SpR}(A)$. Then there exists $\kappa \in [0, \infty)^N$ non zero such that

$$ A\kappa = \rho\kappa. $$

To prove this weak version we only need to consider a sequence $A_\varepsilon \xrightarrow{\varepsilon \to 0} A$ such that $A_\varepsilon$ satisfies the conditions of the classical Perron-Frobenius Theorem. Take for example $A_\varepsilon = A + \varepsilon J$ with $J$ the matrix with only ones. By taking the limit (up to extractions) in

$$ A_\varepsilon \kappa_\varepsilon = \rho_\varepsilon \kappa_\varepsilon $$

we obtain the eigenvector $\kappa \succeq 0$ and again up to extraction we have convergence of the spectrums.

**6.1. Proof of Theorem 3.1.** To prove Theorem 3.1 we introduce several intermediate results. Let us start with the following definition.

**Definition 6.1.** Let $u \in \mathbb{R}_+^M$ and define $\psi_u : x \in \mathbb{R}^M \mapsto u + H(e^x - 1)$. Then we define the increasing limit

$$ \mathcal{L}(u) = \lim_{n \to \infty} \psi_u^n(0) $$

where $\psi_u^n$ is the $n$-fold composition $\psi_u \circ \dots \circ \psi_u$.

The link between this definition of $\mathcal{L}(u)$ and the exponential moments will be done later. For now we stick to this definition and prove the different results on $\mathcal{L}(u)$.

**Proposition 6.2.** Suppose that there exists $x \in \mathbb{R}_+^M$ such that $x = \psi_u(x)$. Then $\mathcal{L}(u)$ is finite and it is the smallest solution (for $\leq$) of the fixed point equation $y = \psi_u(y)$ among the solutions with non negative coordinates. Reciprocally, if $|\mathcal{L}(u)|_\infty < \infty$ then $\mathcal{L}(u)$ is solution of $y = u + H(e^y - 1)$. Finally, the following holds. For any $0 \leq u \leq v$ we have

$$ \mathcal{L}(u) \leq \mathcal{L}(v). $$
For any $u, v \ge 0$ we have

$$ \mathcal{L}(u) + \mathcal{L}(v) \le \mathcal{L}(u + v). $$

**Proof of Proposition 6.2:** Since $H$ has only non negative entries, the function $\psi_u$ is increasing for $\le$. Thus since $u \ge 0$ we have $\mathcal{L}(u) \ge 0$. Let $x$ a non negative and finite fixed point of $\psi_u$. Since $x \ge 0$, applying $\psi_u$ on both side gives $x \ge \psi_u(0)$. By induction, $x \ge \psi_u^n(0)$ for all $n \ge 0$ and thus at the limit, $x \ge \mathcal{L}(u)$ which proves that $\mathcal{L}(u)$ is finite, non negative and smaller than any other solution with non negative coordinates. By continuity of $\psi_u$, it is also clear that $\mathcal{L}(u)$ is a solution of the fixed point equation.

Reciprocally, if $|\mathcal{L}(u)|_\infty < \infty$ it is clear by continuity of $\psi_u$ that $\mathcal{L}(u)$ is a non negative fixed point.

For the last two statements, one can remark that $\psi_u$ and $\psi_v$ are increasing functions for $\le$ and since it is also clear that for $u \le v$ we have $\psi_u \le \psi_v$. Thus by induction,

$$ \psi_u^n(0) \le \psi_v^n(0), \quad n \in \mathbb{N} $$

which leads to $\mathcal{L}(u) \le \mathcal{L}(v)$. Similarly, since $e^{x+y} - 1 \ge e^x - 1 + e^y - 1$ for any $x, y \ge 0$ it is clear that

$$ \psi_{u+v}(x + y) \ge \psi_u(x) + \psi_v(y) $$

for any $x, y \ge 0$. Thus an induction leads to $\psi_{u+v}^n(0) \ge \psi_u^n(0) + \psi_v^n(0)$ which concludes on $\mathcal{L}(u) + \mathcal{L}(v) \le \mathcal{L}(u+v)$. $\square$

From Definition 6.1 and Proposition 6.2 it is now clear that either there exists $x \ge 0$ finite such that $x = u + H(e^x - 1)$ either $|\mathcal{L}(u)|_\infty = \infty$. Thus the following holds

$$ |\mathcal{L}(u)|_\infty < \infty \iff \exists x \in \mathbb{R}_+^M, x = u + H(e^x - 1), \quad (6.1) $$

which is point (1) of Theorem 3.1. It is also clear from Proposition 6.2 that point (2) of Theorem 3.1 holds.

Lemma 3.2 of [11] proves exponential moment for Poissonian Galton Watson processes regardless of the types. The corner stone of the proof of this lemma is the limit behaviour of the recursive sequence $(\psi_{t1}^n(0))_{n \ge 0}$ with $t \ge 0$. The proof concludes that the following holds.

**Lemma 6.3.** Suppose that $H \in \mathbf{Ge}(r, K)$ with $r < 1$. Then for all $0 \le t \le t_0(r, K)$ we have

$$ \mathcal{L}(t1) = \lim_{n \to \infty} \psi_{t1}^n(0) \le t(1 + \frac{2K}{1-r})1, \quad (6.2) $$

with $t_0(r, K)$ defined in Theorem 3.1.

From this result we can easily derive the rest of the proof. Indeed, by Proposition 6.2, Lemma 6.3 and the definition of $t_0(r, K)$ we have for $u \in \mathbb{R}_+^M$ with $|u|_\infty \le t_0(r, K)$ we have

$$ \mathcal{L}(u) \le \mathcal{L}(|u|_\infty 1) \le |u|_\infty (1 + \frac{2K}{1-r}) 1 \le \log (\frac{1+r}{2r}) 1 \quad (6.3) $$

which is (3.7).

Let us continue with the following.

**Proposition 6.4.** Let $u \in \mathbb{R}_+^M$. Then we have (with possibly both side being infinite on some coordinates),

$$ \mathcal{L}(u) \le \sum_{n \ge 0} [H \operatorname{diag}(e^{\mathcal{L}(u)})]^n \times u. \quad (6.4) $$
Thus if we suppose that $\mathrm{SpR}(\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)})) < 1$, we have

$$ \mathcal{L}(u) \preceq (\mathrm{Id} - \mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)}))^{-1} u. \qquad (6.5) $$

In particular if $\mathbf{H} \in \mathbf{Ge}(r, K)$ with $r < 1$ and if $u \in \mathbb{R}_+^M$ with $|u|_\infty \le t_0(r, K)$ then we have

$$ \mathcal{L}(u) \preceq (\mathrm{Id} - \frac{1+r}{2r} \mathbf{H})^{-1} u. \qquad (6.6) $$

**Proof of Proposition 6.4:** Let $u \ge 0$. First one can notice that for $n \ge 1$ we can rewrite $\psi_u^n(0)$ as follows,

$$ \psi_u^n(0) = \sum_{k=0}^{n-1} (\psi_u^{k+1}(0) - \psi_u^k(0)). $$

For $k = 0$, we have $\psi_u^{k+1}(0) - \psi_u^k(0) = u = [\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)})]^0 \times u$. For $k \ge 1$ we have

$$ \begin{aligned} \psi_u^{k+1}(0) - \psi_u^k(0) &= \mathbf{H} (e^{\psi_u^k(0)} - e^{\psi_u^{k-1}(0)}) \\ &\preceq \mathbf{H} \operatorname{diag}(e^{\psi_u^k(0)})(\psi_u^k(0) - \psi_u^{k-1}(0)) \\ &\preceq \mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)})(\psi_u^k(0) - \psi_u^{k-1}(0)) \\ &\preceq [\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)})]^k \times u \end{aligned} $$

where the last inequality is obtained by induction. Thus we have

$$ \mathcal{L}(u) \preceq \sum_{k=0}^{\infty} [\mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)})]^k \times u. $$

And thus under the spectral radius condition, we have

$$ \mathcal{L}(u) = (\mathrm{Id} - \mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)}))^{-1} u. $$

For the last part, if $\mathbf{H} \in \mathbf{Ge}(r, K)$ and $|u|_\infty \le t_0(r, K)$ then by (6.3) $\mathcal{L}(u) \preceq \log(\frac{1+r}{2r})\mathbf{1}$ and so

$$ \mathbf{H} \operatorname{diag}(e^{\mathcal{L}(u)}) \preceq \frac{1+r}{2r} \mathbf{H}. $$

Since $\mathrm{SpR}(\frac{1+r}{2r}\mathbf{H}) \le \frac{1+r}{2} < 1$ the second bound is clear. $\square$

We can now finish the proof of Theorem 3.1. Clearly, (3.5) and (3.8) are Proposition 6.4. Let us now link the exponential moments to Definition 6.1 of $\mathcal{L}$.

Let $u \in \mathbb{R}_+^M$ and $n$ a non negative integer. The branching property of Galton Watson processes leads to the following equality in distribution;

$$ \mathrm{Card}_M(\mathcal{T}_{\le n+1}^{m_0}) = e_{m_0} + \sum_{m \in \mathcal{M}} \sum_{k=1}^{X_{m_0,m}} \mathrm{Card}_M(\mathcal{T}_{\le n}^{m,k}) \qquad (6.7) $$

where $X_{m_0,m}$, $\mathcal{T}_{\le n}^{m,k}$ for $m \in \mathcal{M}$, $k \in \mathbb{N}$ are independent, $X_{m_0,m} \sim \mathrm{Poi}(H_{m_0,m})$ and $\mathcal{T}_{\le n}^{m,k} \sim \mathcal{T}_{\le n}^m$.
Thus if we denote $g_{m_0}^n = \log \mathbb{E}[e^{u \cdot \text{Card}_M(\mathcal{T}_{\le n}^{m_0})}]$ and $g^n = (g_m^n)_{m \in M}$ we have

$$
\begin{align*}
g_{m_0}^{n+1} &= \log \left( \mathbb{E} \left[ e^{u \cdot \boldsymbol{e}_{m_0} + \sum_{m \in M} \sum_{k=1}^{X_{m_0,m}} u \cdot \text{Card}_M(\mathcal{T}_{\le n}^{m,k})} \right] \right) \\
&= u \cdot \boldsymbol{e}_{m_0} + \sum_{m \in M} \log \left( \mathbb{E} \left[ e^{\sum_{k=1}^{X_{m_0,m}} u \cdot \text{Card}_M(\mathcal{T}_{\le n}^{m,k})} \right] \right) \\
&= u \cdot \boldsymbol{e}_{m_0} + \sum_{m \in M} \log \left( \mathbb{E} \left[ \prod_{k=1}^{X_{m_0,m}} \mathbb{E} \left[ e^{u \cdot \text{Card}_M(\mathcal{T}_{\le n}^m)} \right] \right] \right) \\
&= u \cdot \boldsymbol{e}_{m_0} + \sum_{m \in M} \log \left( \mathbb{E} \left[ (e^{g_m^n})^{X_{m_0,m}} \right] \right) \\
&= u \cdot \boldsymbol{e}_{m_0} + \sum_{m \in M} H_{m_0,m} (e^{g_m^n} - 1).
\end{align*}
$$

Thus we have $g^{n+1} = \psi_u(g^n)$. Finally since $g^0 = u = \psi_u(0)$ it is clear by Definition 6.1 that

$$
\lim_{n \to \infty} g^n = \mathcal{L}(u).
$$

Then by monotone convergence theorem, $e_{m_0}^{g_m^n} = \mathbb{E} \left[ e^{u \cdot \text{Card}_M(\mathcal{T}_{\le n}^{m_0})} \right] \xrightarrow{n \to \infty} \mathbb{E} \left[ e^{u \cdot \text{Card}_M(\mathcal{T}^{m_0})} \right]$, which concludes that the definitions of $\mathcal{L}$ by (3.1) and by Definition 6.1 actually coincide. Now that the link between the exponential moments and $\mathcal{L}(u)$ has been done, all the other results are already proved except the convexity property in point (4). From the exponential moments point of view, this inequality is exactly the Hölder inequality, which concludes the proof of Theorem 3.1.

6.2. *Proof of Lemma 3.5.* Clearly, we have

$$
\mathbb{E}[u \cdot \text{Card}_M(\mathcal{T}_0^m)] = u \cdot \boldsymbol{e}_m = (u)_m.
$$

Then, for $n=1$ we have

$$
\mathbb{E}[u \cdot \text{Card}_M(\mathcal{T}_1^m)] = \sum_{m'} H_{m,m'} u_{m'} = (H u)_m.
$$

A classical induction shows that

$$
\mathbb{E}[u \cdot \text{Card}_M(\mathcal{T}_n^m)] = (H^n u)_m, \quad n \ge 0.
$$

Thus we have

$$
\mathbb{E}[u \cdot \text{Card}_M(\mathcal{T}^m)] = e_m \cdot \sum_{n \ge 0} \mathcal{H}^n u
$$

which is the result. From $e^x \ge 1+x$ the lower bound on $\mathcal{L}(u)$ is clear. Let us now assume that $\text{SpR}(\mathcal{H}) \ge 1$ and that $\mathcal{H}$ is irreducible. Let $u \ge 0$ and $u \ne 0$. By irreducibility, there exists $N$ such that

$$
\sum_{k=0}^{N-1} \mathcal{H}^k \succ 0.
$$

Thus is it clear that

$$
\sum_{k=0}^{N-1} \mathcal{H}^k u \succ 0.
$$
Thus we have

$$
\begin{align*}
\mathcal{L}(u) &\succeq \left(\sum_{k=0}^{\infty} \mathbf{H}^k\right) u \\
&\succeq \frac{1}{N}\left(\sum_{k=0}^{\infty} \mathbf{H}^k\right)\left(\sum_{k=0}^{N-1} \mathbf{H}^k u\right) \\
&\succeq \varepsilon\left(\sum_{k=0}^{\infty} \mathbf{H}^k\right)\mathbf{1}
\end{align*}
$$

with some $\varepsilon > 0$. From Perron-Frobenius Theorem, $\rho = \text{SpR}(\mathbf{H}) \ge 1$ is an eigenvalue of $\mathbf{H}$ with an eigenvector $\kappa > 0$. Since we can suppose $\mathbf{1} \succeq \kappa$, we have

$$
\begin{align*}
\mathcal{L}(u) &\succeq \varepsilon \sum_{k=0}^{\infty} \rho^k \kappa \\
&= \infty \cdot \mathbf{1},
\end{align*}
$$

which concludes point (1). Suppose now that there exists $u > 0$ such that $|\mathcal{L}(u)|_\infty < \infty$. Let again, by Weak Perron Frobenius Theorem, $\kappa \succeq 0$ eigenvector of $\mathbf{H}$ such that $\mathbf{H}\kappa = \rho\kappa$ with $\rho = \text{SpR}(\mathbf{H})$. For $\varepsilon > 0$ small enough we have $u \succeq \varepsilon\kappa$ and thus $\mathcal{L}(\varepsilon\kappa)$ is finite. Considering the bound

$$
\mathcal{L}(\varepsilon\kappa) \succeq \sum_{n \ge 0} \mathbf{H}^n \varepsilon\kappa = \varepsilon \left( \sum_{n \ge 0} \rho^n \right) \kappa,
$$

we must have $\rho < 1$, which is point (2).

6.3. *Proof of Theorem 3.6.* For $u \ge 0$ consider the real function $\phi_u(x) = u + \alpha(e^x - 1) - x$. We want to find for which $u$ this function has non negative zeros. $\phi_u$ is strictly convexe, $\phi_u(0) = u \ge 0$, $\phi_u'(0) < 0$ and $\lim_{x \to \infty} \phi_u(x) = \infty$ thus if $\phi_u$ has zeros they must be non negative. A quick study of $\phi_u$ shows that its minimum is reached at $x = \log(1/\alpha)$. Thus, if $\min \phi_u > 0$ there are no zeros, if $\min \phi_u = 0$ there is 1 zero and finally if $\min \phi_u < 0$ there are 2 zeros. Since for a fixed $x$ the quantity $\phi_u(x)$ is increasing with $u$, it is clear that the largest $u$ such that there exists at least one zero is the one such that

$$
0 = \min \phi_u = \phi_u(\log(1/\alpha)) = u + (1 - \alpha) - \log(1/\alpha).
$$

Which gives $u = u_c = \log(1/\alpha) - (1-\alpha) > 0$ and $x_c = \mathcal{L}(u_c) = \log(1/\alpha)$.

The last statement is just that $\mathcal{L}(u)$ is the smallest non negative zero, and thus at $x = \mathcal{L}(u)$ we
must have, by convexity, $\phi'_u(x) \le 0$.

6.4. *Proof of Theorem 3.7.* Recall that the clusters of the Hawkes process are homogeneous *Pois*(h) clusters, i.e. $h_{m,m'}(s,t) = h_{m,m'}(t-s)$, and we also have for Hawkes processes, $h_{m,m'}(s) = 0$ for $s < 0$. Each cluster consists in a structure of *Pois*(H) Galton Watson processes with an embedding on the real line: the birth dates of each point, see [11].

Just by following the steps of [11] we have that for $u \in \mathbb{R}_+^M$,

$$
\mathbb{E} \left[ e^{u \cdot \mathbf{N}([0,L])} \right] = \exp \left( \sum_{m \in \mathcal{M}} \int_{-L}^{\infty} [\psi(s, u, m, L) - 1] \mu_m ds \right) \quad (6.8)
$$

where

$$
\psi(s, u, m, L) = \mathbb{E} \left[ e^{u \cdot \text{Card}_M \left( G_0^m \cap [s, s+L) \right)} \right] \quad (6.9)
$$

where $G_0^m$ is a $\mathcal{P}$ois($\mathbf{h}$) cluster with root of type $m$ and born at time 0.
Using the same argument as in [11], in addition with Theorem 3.1, lead to the following for $-L < s \le 0$,

$$ \sum_{n=0}^{\infty} [\psi(s + nL, u, m, L) - 1] \le \mathbb{E} [e^{u \cdot \text{Card}_M(G_0^m \cap \mathbb{R})}] - 1 = \exp(e_m \cdot \mathcal{L}(u)) - 1, \quad (6.10) $$

with $\mathcal{L}(u)$ from Theorem 3.1. Thus we have

$$ \begin{aligned} \mathbb{E} [e^{u \cdot \mathcal{N}([0,L])}] &= \exp \left( \sum_{m \in \mathcal{M}} \int_{-L}^{\infty} [\psi(s, u, m, L) - 1] \mu_m ds \right) \\ &= \exp \left( \sum_{m \in \mathcal{M}} \int_{-L}^{0} \sum_{n \ge 0} [\psi(s + nL, u, m, L) - 1] \mu_m ds \right) \\ &\le \exp \left( \sum_{m \in \mathcal{M}} \mu_m L (e^{e_m \cdot \mathcal{L}(u)} - 1) \right) \\ &= \exp \left( L \mu \cdot (e^{\mathcal{L}(u)} - 1) \right). \end{aligned} $$

Finally, Theorem 3.1 concludes by giving the bound on $\mathcal{L}(u)$ if $H \in \mathbf{Ge}(r, K)$, and $|u|_{\infty} \le t_0(r, K)$.

6.5. *Proof of Theorem 3.10.* Theorem 3.10 is a direct application of Theorem 4.5 as explained below the latter. Indeed, one just has to consider $h = \delta_1 H$ and apply Theorem 4.5. Equation (4.3) gives the fixed point equation and point (3) the bound.

6.6. *Proof of Theorem 4.1.* Let $u \in \mathbb{R}_+^M$ and $G_s^m$ an inhomogeneous $\mathcal{Poi}(h)$ cluster with root of type $m$ and born at time $s$. Recall that for $m \in \mathcal{M}$, $s \in \mathbb{R}$ and $t \in \mathbb{R}$ the function $f_{u,m}$ is defined by

$$ f_{u,m}(s,t) = \log \left( \mathbb{E} \left[ e^{u \cdot \text{Card}_M (G_s^m \cap [t, \infty))} \right] \right). $$

We can use the branching property on the first generation as follows,

$$ e^{f_{u,m}(s,t)} = \mathbb{E} \left[ e^{u \cdot \text{Card}_M (G_s^m \cap [t, \infty))} \right] = \mathbb{E} \left[ e^{u \cdot e_m 1_{t \le s}} \prod_{m' \in \mathcal{M}} \prod_{k=1}^{X_{m,m'}(s)} e^{u \cdot \text{Card}_M (G_{Z^{m'},k}^{m'} \cap [t, \infty))} \right] $$

where $X_{m,m'}(s) \sim \mathcal{Poi}(\|h_{m,m'}(s, \cdot)\|_1)$, and $Z^{m',k} \sim Z^m \sim \frac{h_{m,m'}(s,t)}{\|h_{m,m'}(s, \cdot)\|_1} ds$ and the X's and the Z's are all independent. Thus we have

$$ \begin{aligned} e^{f_{u,m}(s,t)} &= \mathbb{E} \left[ e^{u \cdot e_m 1_{t \le s}} \prod_{m' \in \mathcal{M}} \prod_{k=1}^{X_{m,m'}(s)} \mathbb{E}_{Z^{m'}} \left[ e^{f_{u,m'}(Z^{m'},t)} \right] \right] \\ &= \exp \left[ u \cdot e_m 1_{t \le s} + \sum_{m'} \int_{\mathbb{R}} h_{m,m'}(s,w) \times (e^{f_{u,m'}(w,t)} - 1) dw \right] \\ &= \exp \left[ \left( u 1_{t \le s} + [h \star (e^{f_u} - 1)](s,t) \right)_m \right]. \end{aligned} $$

Thus we have

$$ f_u(s,t) = u1_{t \le s} + [h \star (e^{f_u} - 1)](s,t) $$

which proves (4.1). Let us now prove (4.2). Define $f_u^1$ by

$$ f_u^1(s,t) = u1_{t \le s} $$
and for $n \ge 2$ let $f_u^n$ defined by

$$f_u^n(s,t) = u1_{t \le s} + [h \star (e^{f_u^{n-1}} - 1)](s,t). \quad (6.11)$$

Then, as is the proof of Theorem 3.1, one can check that $f_u^n$ is obtained by replacing the clusters by clusters caped after generation $n-1$ in the definition of $f_u$. Thus by monotone convergence theorem, $f_u^n(s,t) \xrightarrow{n \to \infty} f_u(s,t)$ for any $s,t \in \mathbb{R}$. If we let $f_u^0(s,t) = 0$ then (6.11) also holds for $n=1$. Thus we may write

$$f_u = \sum_{n \ge 0} (f_u^{n+1} - f_u^n).$$

Since the underlying $M$-type tree of a cluster is stochastically dominated by a $\mathcal{P}$oits($H$) Galton Watson tree with $H = \|h\|_{L^\infty L^1}$ it is clear that $f_u^n(s,t) \le \mathcal{L}(u)$ for any $n \ge 0$ and any $s,t \in \mathbb{R}$. Thus, for any $n \ge 1$ we have

$$\begin{aligned} f_u^{n+1} - f_u^n &= h \star (e^{f_u^n} - e^{f_u^{n-1}}) \\ &\le h \operatorname{diag}(e^{\mathcal{L}(u)}) \star (f_u^n - f_u^{n-1}) \\ &\le (h \operatorname{diag}(e^{\mathcal{L}(u)}))^{*n} \star (f_u^1 - f_u^0) \\ &= (h \operatorname{diag}(e^{\mathcal{L}(u)}))^{*n} \star u1_{D_{-}}. \end{aligned}$$

Since $f_u^1 - f_u^0 = u1_{D_{-}}$ we have

$$\begin{aligned} f_u &= \sum_{n \ge 0} (f_u^{n+1} - f_u^n) \\ &\le u1_{D_{-}} + \sum_{n \ge 1} (h \operatorname{diag}(e^{\mathcal{L}(u)}))^{*n} \star u1_{D_{-}} \\ &= u1_{D_{-}} + \Psi_u \star 1_{D_{-}} \operatorname{Id} \times u \\ &= u1_{D_{-}} + R_{\Psi_u} \times u, \end{aligned}$$

which proves (4.2).

Finally let us prove points (1), (2) and (3). Points (1) and (2) are straightforward from Proposition A.8 and Proposition A.7 respectively. Indeed, in the case where $H \in \mathbf{Ge}(r, K)$ and $|u|_\infty \le t_0(r, K)$, we have that $|\mathcal{L}(u)|_\infty \le \log((1+r)/2r)$ and thus it is clear that $h \operatorname{diag}(e^{\mathcal{L}(u)}) \in \mathbf{Ge}((1+r)/2, K)$ which concludes by the above mentioned Propositions. Let us now prove point (3).

Let for $n \in \mathbb{N}$,

$$g_u(n) = \|s \in \mathbb{R} \mapsto f_u(s, s + nA)\|_{L^\infty}.$$

Recall that $f_u(s, \cdot)$ is decreasing. Thus, since $h$ is supported in $\{(s,t) \in \mathbb{R}^2 | t \le s + A\}$, we have for $n \in \mathbb{N}^*$

$$\begin{aligned} f_u(s, s + nA) &= \int_{-\infty}^{s+A} h(s, w)(e^{f_u(w, s+nA)} - 1)dw \\ &\le \int_{-\infty}^{s+A} h(s, w)(e^{f_u(w, w+(n-1)A)} - 1)dw \\ &\le \int_{-\infty}^{s+A} h(s, w)(e^{g_u(n-1)} - 1)dw \\ &\le H(e^{g_u(n-1)} - 1). \end{aligned}$$
Thus we have

$$ \forall n \in \mathbb{N}, \quad g_u(n+1) \preceq H(e^{g_u(n)} - 1), \quad \text{and} \quad g_u(0) \preceq \mathcal{L}(u). $$

By induction one has, for all $n \ge 0$,

$$ g_u(n) \preceq e^{\sum_{k=0}^{n-1} |g_u(k)|_\infty} H^n \mathcal{L}(u). \quad (6.12) $$

Since $\mathcal{L}(u)$ is finite, it is clear that $e^{u \cdot \text{Card}_M(T^m)}$ is integrable where $T^m$ is a $\mathcal{P}$oïd$(H)$ Galton Watson tree. By definition of $H = \|h\|_{L^\infty L^1}$ and by the support assumption, it is clear that for any $s \in \mathbb{R}$ and any $n \in \mathbb{N}$, the random variables $e^{u \cdot \text{Card}_M(G_s^m \cap [s+nA, \infty))}$ are all stochastically dominated by $e^{u \cdot \text{Card}_M(T_{\ge n}^m)}$. Thus by dominated convergence theorem we have

$$ g_u(n) \xrightarrow{n \to \infty} 0. $$

Thus for any $\varepsilon > 0$ there exists $C = C_\varepsilon$ (depending on $u$ and $H$ and $A$) such that

$$ \sum_{k=0}^{n-1} |g_u(k)|_\infty \le \varepsilon n + C_\varepsilon, \quad n \ge 0. $$

Choose $\varepsilon > 0$ small enough so that $\text{SpR}(H) < e^{-\varepsilon}$. Since $\|H^n\|_\infty^{1/n} \xrightarrow{n \to \infty} \text{SpR}(H)$ we have

$$ |g_u(n)|_\infty \le e^{C_\varepsilon \varepsilon n} \|H^n\|_\infty |\mathcal{L}(u)|_\infty \\ \le C \alpha^n $$

for some $C$ and $\alpha < 1$ depending on $u$ and $H$ and $A$. With the new bound we have

$$ \sum_{k=0}^{\infty} |g_u(k)|_{\infty} < \infty, $$

which proves point (3) in the first case by plugging in (6.12).

Finally, suppose that $H \in \mathbf{Ge}(r, K)$ with $r < 1$ and let $u \in \mathbb{R}_+^M$ such that $|u|_\infty \le t_0(r, K)$. By Theorem 3.1, one has $|g_u(n)|_\infty \le |\mathcal{L}(u)|_\infty \le \log(\frac{1+r}{2r})$ for any $n \ge 0$. By (6.12) one has,

$$ g_u(n) \preceq \left(\frac{1+r}{2r}\right)^n \mathcal{L}(u). $$

Thus since $H \in \mathbf{Ge}(r, K)$ we have for $n \ge 1$,

$$ |g_u(n)|_\infty \le K \left(\frac{1+r}{2}\right)^n \log \left(\frac{1+r}{2r}\right). $$

Plugging this new inequality in (6.12) leads to

$$ \begin{aligned} g_u(n) &\preceq \left(\frac{1+r}{2r}\right)^{1+K \sum_{k=1}^{n-1} \left(\frac{1+r}{2}\right)^k} H^n \mathcal{L}(u) \\ &\preceq \left(\frac{1+r}{2r}\right)^{1+2K/(1-r)} H^n \mathcal{L}(u), \end{aligned} $$

which is the result. Theorem 4.1 is proved.
6.7. *Proof of Theorem 5.6. (1) :* By (3.4) of Theorem 3.1 it is clear that $E$ is convex and that $E = \text{LC}_{\leq}(E) \cap \mathbb{R}_+^M$. Let now $u_n \in E$ such that $u_n \to u^*$. Thus we have

$$u_n = \mathcal{L}(u_n) - \mathbf{H}(e^{\mathcal{L}(u_n)} - \mathbf{1}) \xrightarrow{n \to \infty} u^*.$$

Suppose that there exists $S \subset M$ and an extraction $\phi(n)$ such that $\forall i \in S$ we have $\mathcal{L}(u_{\phi(n)})_i \to \infty$ and $\mathcal{L}(u_{\phi(n)})_i$ is bounded for $i \notin S$. By extracting we can suppose that there is $i_0$ in $S$ such that, for $n$ large enough,

$$\min_{i \in S} \mathcal{L}(u_{\phi(n)})_i = \mathcal{L}(u_{\phi(n)})_{i_0}.$$

Then, either $H_{i_0,j} = 0$ for all $j \in S$ and we have $u_{i_0}^* = \infty$ either there exists $j \in S$ such that $H_{i_0,j} > 0$ and in this case $u_{i_0}^* = -\infty$. In both cases we have a contradiction thus $\mathcal{L}(u_n)$ is bounded. Thus, up to extraction we can assume that

$$\mathcal{L}(u_n) \xrightarrow{n \to \infty} y.$$

Clearly by continuity we have $y = u^* + \mathbf{H}(e^y - \mathbf{1})$ and $y \geq 0$ finite and thus $u^* \in E$ by Theorem 3.1. Thus $E$ is closed.

(2) : Lemma 3.5 gives the direct implication and by Theorem 3.1 the reciprocal is also clear. Let us prove that $\hat{E}$ is indeed the interior of $E$ if $\text{SpR}(\mathbf{H}) < 1$. Suppose that $\text{SpR}(\mathbf{H}) < 1$ and let $u \in \hat{E}$. Then there exists $\varepsilon > 0$ such that $\varepsilon \mathbf{1}$ and $(1+\varepsilon)u$ are in $E$. By convexity we have

$$u + \frac{\varepsilon^2}{1 + \varepsilon} \mathbf{1} \in E.$$

Thus it is clear that the ball, in $\mathbb{R}_+^M$, centered at $u$ with radius $\frac{\varepsilon^2}{1+\varepsilon}$ for the norm $|\cdot|_\infty$ lies entirely in $E$. Reciprocally, if $u \in \delta E$ then it is clear that by definition that $u$ is not in the interior, which concludes since $E$ is closed in $\mathbb{R}^M$ (and thus in $\mathbb{R}_+^M$).

(3) : Let $m \in M$, and suppose that $\text{SpR}(\mathbf{H}_{\|C(m)}) < 1$. Then from Lemma 5.5 and point (2) it is clear that $\varepsilon e_m \in E$ for $\varepsilon > 0$ small enough. Reciprocally, suppose that $\text{SpR}(\mathbf{H}_{\|C(m)}) \geq 1$. Let $m \in M$, and $\varepsilon > 0$. By Lemma 3.5 we have

$$\mathcal{L}(\varepsilon e_m) \succeq \varepsilon \sum_{n \geq 0} \mathbf{H}^n e_m.$$

Denote $z = (e_m)_{|C(m)}$ and $A = \mathbf{H}_{\|C(m)}$, thus from the support considerations of Lemma 5.5 and Remark 5.4 we have,

$$(\mathcal{L}(\varepsilon e_m))_{|C(m)} \succeq \varepsilon \sum_{n \geq 0} A^n z.$$

Let $\mathbf{1}_{C(m)} = (1, \dots, 1)^\top \in \mathbb{R}^{C(m)}$. By definition of $C(m)$, there exist $N \geq 1$ and $\eta > 0$ such that we have $\sum_{k=0}^{N-1} A^k z \geq \eta \mathbf{1}_{C(m)} > 0$ in $\mathbb{R}^{C(m)}$. Thus we have,

$$\begin{aligned} (\mathcal{L}(\varepsilon e_m))_{|C(m)} &\succeq \varepsilon N^{-1} \sum_{n \geq 0} A^n \sum_{k=0}^{N-1} A^k z \\ &\succeq \varepsilon \eta N^{-1} \sum_{n \geq 0} A^n \mathbf{1}_{C(m)}. \end{aligned}$$
By assumption, $\rho = \text{SpR}(A) \ge 1$, thus by Weak Perron-Frobenius Theorem there exists a non zero $\kappa \succeq 0$ such that $A\kappa = \rho\kappa$. Without loss of generality we can suppose $1_{C(m)} \succeq \kappa$. Thus we have

$$ (\mathcal{L}(\varepsilon e_m))_{|C(m)} \succeq \varepsilon\eta N^{-1} \sum_{n \ge 0} \rho^n \kappa = \infty\kappa. $$

Which proves that $\varepsilon e_m \notin E$, where $\varepsilon > 0$ is arbitrary, which concludes.

Let us now prove the second equivalence. On one hand, if $E = \{0\}$ then for any $m \in M$ and any $\varepsilon > 0$ we have $\varepsilon e_m \notin E$ and thus $\text{SpR}(H_{\parallel C(m)}) \ge 1$. Reciprocally, for all $m \in M$, for all $\varepsilon > 0$ we have $\varepsilon e_m \notin E$. Thus if $E \ne \{0\}$ we would have a contradiction since $u \ne 0$ in $E$ has at least one positive coordinate, say $m$, and thus $u_m e_m \in E$ which is absurd.

(4) : Let $u, v \in E$ such that $\mathcal{L}(u) = \mathcal{L}(v)$. Then it is clear that $u = \mathcal{L}(u) - H(e^{\mathcal{L}(u)} - 1) = v$ which proves that $\mathcal{L}$ is bijective onto its image. One can extend $\mathcal{L}$ as follows,

$$ \mathcal{L}: \mathrm{LC}_{\le}(E) \longrightarrow \mathbb{R}^M \\ u \longmapsto \left[ \log \left( \mathrm{E} \left[ e^{u \cdot \mathrm{Card}_M(\mathcal{T}^m)} \right] \right) \right]_{m \in M}. $$

Continuity on $\mathrm{LC}_{\le}(\{u\})$ is clear by dominated convergence, which also gives continuity on $\hat{E}$. Suppose now that $\text{SpR}(H) < 1$. Thus $\mathrm{LC}_{\le}(\hat{E})$ is an open set of $\mathbb{R}^M$. For any $u \in \mathrm{LC}_{\le}(\hat{E})$ there exists a neighbourhood $V$ of $u$, $\varepsilon > 0$ and $u_V \in \hat{E}$ such that for all $v \in V$ we have $v \preceq u_V - \varepsilon \mathbf{1}$. Thus from classical results on integral depending on a parameter, it is clear that it is infinitely differentiable since one can apply dominated convergence theorems to all the derivatives.

(5) : Suppose that there is two such solutions $x, y$. Clearly $u \in E$ and thus we will show that $y = z = \mathcal{L}(u)$. Thus we can suppose that $y = \mathcal{L}(u) \preceq z$ since $\mathcal{L}(u)$ is smaller than any other solution. We have

$$ \begin{aligned} z - \mathcal{L}(u) &= H(e^z - e^{\mathcal{L}(u)}) \\ &\preceq H \operatorname{diag}(e^z)(z - \mathcal{L}(u)) \\ &\preceq \limsup_{n \to \infty} [H \operatorname{diag}(e^z)]^n (z - \mathcal{L}(u)). \end{aligned} $$

Thus if $\text{SpR}(H \operatorname{diag}(e^z)) < 1$ the result is clear.

If $\text{SpR}(H \operatorname{diag}(e^z)) = 1$ then the argument is more subtle. Denote $x = z - \mathcal{L}(u) \succeq 0$. We have

$$ x = H \operatorname{diag}(e^{\mathcal{L}(u)})(e^x - \mathbf{1}). $$

Let $S \subset M$ the support of $x$ and $A = H \operatorname{diag}(e^{\mathcal{L}(u)})$. Because the support of $e^x - \mathbf{1}$ is also $S$ it is clear that for all $i \in S$ there is at least one $j \in S$ such that $A_{ij} > 0$ and that $A_{ij} = 0$ for any $i \notin S$ and $j \in S$. Let $\psi$ the function defined by

$$ \Phi: \mathbb{R} \longrightarrow \mathbb{R}^M \\ t \longmapsto A(e^{tx} - \mathbf{1}) - tx. $$

Each coordinate of $\Phi$ is a convex function (as non negative linear combination of convex functions). More precisely, from the support considerations, it is clear that for all $i \in S$ the $i$-th coordinate of $\Phi$ is strictly convexe since there exists $j \in S$ such that $A_{ij} > 0$ and $t \mapsto e^{tx_j} - 1$ is strictly convex. Also, the $i$-th coordinate of $\Phi$ for $i \notin S$ is constant of value 0. Thus, since $\Phi(0) = \Phi(1) = 0$ we must have for some $\varepsilon > 0$ is small enough,

$$ \Phi'(1) = A \operatorname{diag}(e^x)x - x \succeq \varepsilon \mathbf{1}_S $$
where $\mathbf{1}_S$ is the vector with coordinates 1 on elements of $S$ and 0 outside. Thus up to reducing $\varepsilon$ we may assume that

$$A \operatorname{diag}(e^x)x - x \succeq \varepsilon x.$$

Thus we have

$$x \preceq (1 + \varepsilon)^{-1} A \operatorname{diag}(e^x)x.$$

One can iterate and obtains

$$x \preceq ((1 + \varepsilon)^{-1} A \operatorname{diag}(e^x))^{n} x, \quad \forall n \geq 0.$$

Recall that $A \operatorname{diag}(e^x) = H \operatorname{diag}(e^z)$ is assumed to have spectral radius equal to one. Thus $x = 0$ and $z = \mathcal{L}(u)$.

(6): First remark that both set on the right hand side are equal by (5). Let $u \in \hat{E}$. It is clear from the branching property that

$$\forall v \in \mathrm{LC}_{\preceq}(\hat{E}), v = \mathcal{L}(v) - H(e^{\mathcal{L}(v)} - \mathbf{1}).$$

Thus, since $u$ is in the open (for $\mathbb{R}^M$) set $\mathrm{LC}_{\preceq}(\hat{E})$ one can differentiate the equality $u = \mathcal{L}(u) - H(e^{\mathcal{L}(u)} - \mathbf{1})$ and obtains

$$\mathrm{Id} = (\mathrm{Id} - H \operatorname{diag}(e^{\mathcal{L}(u)})) D_u \mathcal{L}. \qquad (6.13)$$

Thus, $(\mathrm{Id} - H \operatorname{diag}(e^{\mathcal{L}(u)}))^{-1}$ exists. Suppose that $\rho := \mathrm{SpR}(H \operatorname{diag}(e^{\mathcal{L}(u)})) = 1$. Then since $H \operatorname{diag}(e^{\mathcal{L}(u)}) \succeq 0$ by the Weak Perron-Frobenius theorem, 1 is eigenvalue, which is absurd. Suppose then that $\rho > 1$. Since $\mathrm{SpR}(H) < 1$, by continuity, there exists $0 \preceq v \preceq u$ such that $\mathrm{SpR}(H \operatorname{diag}(e^{\mathcal{L}(v)})) = 1$ which is absurd since $v \in \hat{E}$ as we shown in the first case. Thus $\rho < 1$, which proves the inclusion "$\subset$".

Let $u \in E$ such that $\mathrm{SpR}(H e^{\mathcal{L}(u)}) < 1$. Let $\varepsilon > 0$ and $v = u + \varepsilon \mathbf{1}$ and define $z_n$ by

$$\forall n \in \mathbb{N}, z_n = \psi_v^n(0) - \psi_u^n(0) \succeq 0,$$

where $\psi_u(x) = u + H(e^x - \mathbf{1})$. Then for $n \geq 0$ we have

$$\begin{aligned} z_{n+1} &= \varepsilon \mathbf{1} + H(e^{\psi_v^n(0)} - e^{\psi_u^n(0)}) \\ &\preceq \varepsilon \mathbf{1} + H \operatorname{diag}(e^{\psi_u^n(0)})(e^{z_n} - \mathbf{1}) \\ &\preceq \varepsilon \mathbf{1} + H \operatorname{diag}(e^{\mathcal{L}(u)})(e^{z_n} - \mathbf{1}). \end{aligned}$$

Thus we have

$$\begin{aligned} z_{n+1} &\preceq g_{\varepsilon \mathbf{1}}(z_n) \\ &\preceq g_{\varepsilon \mathbf{1}}^{n+1}(z_0) \\ &= g_{\varepsilon \mathbf{1}}^{n+1}(0) \end{aligned}$$

where $g_{\varepsilon \mathbf{1}}(x) = \varepsilon \mathbf{1} + H \operatorname{diag}(e^{\mathcal{L}(u)})(e^x - \mathbf{1})$. Since $H \operatorname{diag}(e^{\mathcal{L}(u)})$ has spectral radius smaller than one, it is clear by (2) applied to $H \leftarrow H \operatorname{diag}(e^{\mathcal{L}(u)})$ that for $\varepsilon$ small enough $(z_n)_n$ is bounded and thus converges to a finite limit: indeed, $(\psi_u^n(0))_n$ converges and $(\psi_v^n(0))_n$ is increasing and now bounded thus converges to a finite limit. Which proves that $v \in E$ and thus $u \in \hat{E}$.

(7): First remark that both set on the right hand side are equal by (5).

Let $u \in \delta E$. Then for any $t < 1$ we have $tu \in \hat{E}$ and thus $\mathrm{SpR}(H \operatorname{diag}(e^{\mathcal{L}(tu)})) < 1$. By continuity on $\mathrm{LC}_{\preceq}(\{u\})$ we have $\mathrm{SpR}(H \operatorname{diag}(e^{\mathcal{L}(u)})) \leq 1$. Since $u \notin \hat{E}$, by (6) we must have
SpR($\mathbf{H}$ diag($e^{\mathcal{L}(u)}$)) = 1.

Let $u \in E$ such that SpR($\mathbf{H}$ diag($e^{\mathcal{L}(u)}$)) = 1. Then by (6) it is clear that $u \notin \hat{E}$, which concludes.

Theorem 5.6 is proved.

## Appendix A. Technical results on bivariate convolution

In this section we introduce some results about bivariate convolution which are key to prove Theorem 4.1.

Let $\delta^0 : (s,t) \mapsto \delta_0(t-s)$ with $\delta_0$ the classical Dirac mass at 0 so that $\delta^0$ is the unit for the bivariate convolution $\star$.

This first result extends the classical $L^\infty$ and $L^1$ bounds of a convolution to bivariate convolution.

**Proposition A.1.** Let $\mathbf{v} : \mathbb{R}^2 \to \mathcal{M}_{n,p}$ and $\mathbf{w} : \mathbb{R}^2 \to \mathcal{M}_{p,q}$ bivariate matrix functions. The following holds,

* For any $s \in \mathbb{R}$ we have

$$ \begin{gathered} \|\mathbf{v} \star \mathbf{w}(s, \cdot)\|_{L^1} \le \|\mathbf{v}(s, \cdot)\|_{L^1} \times \|\mathbf{w}\|_{L^\infty L^1}. \\ \text{Thus we have} \\ \|\mathbf{v} \star \mathbf{w}\|_{L^\infty L^1} \le \|\mathbf{v}\|_{L^\infty L^1} \times \|\mathbf{w}\|_{L^\infty L^1}. \end{gathered} $$

* For any $s, t \in \mathbb{R}$ we have

$$ \begin{gathered} |\mathbf{v} \star \mathbf{w}(s, t)| \le \|\mathbf{v}(s, \cdot)\|_{L^1} \times \|\mathbf{w}(\cdot, t)\|_{L^\infty}. \\ \text{Thus we have} \\ \|\mathbf{v} \star \mathbf{w}\|_{L^\infty L^\infty} \le \|\mathbf{v}\|_{L^\infty L^1} \times \|\mathbf{w}\|_{L^\infty L^\infty}. \end{gathered} $$

*Remark A.2.* If $\mathbf{v}(s, t) = f(t-s)$ and $\mathbf{w}(s, t) = g(t-s)$ then this result is just the classical result $\|f \star g\|_{L^1} \le \|f\|_{L^1} \|g\|_{L^1}$ and $\|f \star g\|_{L^\infty} \le \|f\|_{L^\infty} \|g\|_{L^\infty}$.

The following Proposition links the decay of $\mathbf{v}$ and $\mathbf{w}$ to the decay of $\mathbf{v} \star \mathbf{w}$.

**Proposition A.3.** Let $\mathbf{v} : \mathbb{R}^2 \to \mathcal{M}_{n,l}$ and $\mathbf{w} : \mathbb{R}^2 \to \mathcal{M}_{l,r}$ non negative (entry by entry) bivariate matrix functions. Then for any $d \ge 0$ and any $p \in [0, 1]$ the following holds

$$ R_{\mathbf{v} \star \mathbf{w}}^{\infty}(d) \le \|\mathbf{v}\|_{L^{\infty} L^1} \times R_{\mathbf{w}}^{\infty}(qd) + R_{\mathbf{v}}^{\infty}(pd) \times \|\mathbf{w}\|_{L^{\infty} L^1} \quad (A.1) $$

with $q = 1-p$.

*Remark A.4.* This result extends to functions $\mathbf{v}$ and $\mathbf{w}$ not necessarily non negative by replacing $R_{\mathbf{v}}^{\infty}$ and $R_{\mathbf{w}}^{\infty}$ by $R_{|\mathbf{v}|}^{\infty}$ and $R_{|\mathbf{w}|}^{\infty}$.

This result is the starting point for the following proposition that links the decay of a non negative bivariate matrix function $\mathbf{v}$ to the decay of $\psi = \sum_{n \ge 1} \mathbf{v}^{*n}$.

**Proposition A.5.** Let $\mathbf{v}$ a bivariate matrix function such that $\mathbf{v}$ is non negative (entry by entry). Suppose that $\mathbf{V} = \|\mathbf{v}\|_{L^\infty L^1}$ has spectral radius smaller than one then the function $\psi = \sum_{n \ge 1} \mathbf{v}^{*n}$ is well defined and satisfies for all $d > 0$ and all $p \in (0, 1)$,

$$ R_{\psi}^{\infty}(d) \le \sum_{k=0}^{\infty} \mathbf{V}^{k} R_{\mathbf{v}}^{\infty}((1-p)p^{k}d)(\mathrm{Id}-\mathbf{V})^{-1}. $$
Remark A.6. Once again this result applies for $\mathbf{v}$ possibly not non negative by using $|\mathbf{v}|$ instead of $\mathbf{v}$ in the upper bound.

**Proposition A.7.** Let $\mathbf{v}$ non negative (entry by entry) and satisfying $\mathbf{V} = \|\mathbf{v}\|_{L^\infty L^1} \in \mathbf{Ge}(\alpha, K)$ with $\alpha < 1$ and such that

$$R_{\mathbf{v}}^{\infty}(t) \leq \frac{A}{(1+t)^{\gamma}} \mathbf{V}, \quad t \geq 0.$$

Then the function $\psi = \sum_{n \geq 1} \mathbf{v}^{*n}$ is well defined and satisfies for $t \geq 1$ and any $0 < \delta < 1$

$$R_{\psi}^{\infty}(t) \leq \frac{A(1 - e^{-\delta \log(1/\alpha)/\gamma})^{-\gamma}}{(1+t)^{\gamma(1-\delta)}} \mathbf{V}(\mathrm{Id} - \mathbf{V})^{-2} + \mathbf{V}^{\lfloor \frac{\gamma \log(t)}{(1+\delta)\log(1/\alpha)} \rfloor + 2}(\mathrm{Id} - \mathbf{V})^{-2}.$$

In particular,

$$\| R_{\psi}^{\infty}(t) \|_{\infty} \leq \frac{\alpha K(K+1) \left[ A \left( 1 + \frac{\gamma}{\delta \log(1/\alpha)} \right)^{\gamma} + 2^{\gamma} \right]}{(1-\alpha)^2 (1+t)^{\gamma(1-\delta)}}, \quad t \geq 1.$$

**Proposition A.8.** Let $\mathbf{v}$ non negative (entry by entry) such that $\mathbf{V} = \|\mathbf{v}\|_{L^\infty L^1} \in \mathbf{Ge}(\alpha, K)$ with $\alpha < 1$ and that there exists $A \geq 0$ and $c > 0$ such that

$$R_{\mathbf{v}}^{\infty}(t) \leq A e^{-ct} \mathbf{V}, \quad t \geq 0.$$

Then the function $\psi = \sum_{n \geq 1} \mathbf{v}^{*n}$ is well defined and satisfies for all $t > 0$ and all $0 < \varepsilon < \frac{c(1-\alpha)}{(1-\alpha)+A\alpha}$,

$$R_{\psi}^{\infty}(t) \leq e^{-\varepsilon t} \mathbf{V}_{\varepsilon}(\mathrm{Id} - \mathbf{V}_{\varepsilon})^{-1}$$

with

$$\mathbf{V}_{\varepsilon} \leq \alpha \left(1 + \frac{A\varepsilon}{c - \varepsilon}\right) \mathbf{V}.$$

In particular,

$$\| R_{\psi}^{\infty}(t) \|_{\infty} \leq \frac{K \alpha \left(1 + \frac{A\varepsilon}{c-\varepsilon}\right)}{1 - \alpha \left(1 + \frac{A\varepsilon}{c-\varepsilon}\right)} e^{-\varepsilon t}.$$

Remark A.9. In the one dimensional case, if one take $v(s, t) = ae^{-b(t-s)}1_{s \leq t}$ with $b > a > 0$, then in the proof of Lemma 9 of [15] the authors have shown that

$$\psi(s,t) = ae^{-(b-a)(t-s)}1_{s \leq t},$$

which proves that

$$R_{\psi}^{\infty}(t) \lesssim e^{-(b-a)t}.$$

This choice for $\mathbf{v}$ leads to $R_{\mathbf{v}}^{\infty}(t) = \frac{a}{b}e^{-bt} = 1 \times e^{-bt}V$ with $V = a/b \in \mathbf{Ge}(a/b, 1)$. Thus Proposition A.8 gives the following limiting value for $\varepsilon$,

$$\frac{b(1 - a/b)}{(1 - a/b) + 1 \times a/b} = b - a.$$
A.1. *Proof of Appendix A results.*

*Proof of Proposition A.1:* Let $s \in \mathbb{R}$, then

$$ 
\begin{aligned}
\int |\boldsymbol{v} \star \boldsymbol{w}(s,t)|dt &= \int \left| \int \boldsymbol{v}(s,x)\boldsymbol{w}(x,t)dx \right|dt \\
&\leq \int \int |\boldsymbol{v}(s,x)||\boldsymbol{w}(x,t)|dtdx \\
&\leq \int \left( |\boldsymbol{v}(s,x)| \int |\boldsymbol{w}(x,t)|dt \right)dx \\
&\leq \int |\boldsymbol{v}(s,x)|dx\|\boldsymbol{w}\|_{L^\infty L^1} \\
&\leq \|\boldsymbol{v}\|_{L^\infty L^1} \|\boldsymbol{w}\|_{L^\infty L^1}.
\end{aligned}
$$

Similarly, for any $s, t \in \mathbb{R}$ we have

$$ 
\begin{aligned}
|\boldsymbol{v} \star \boldsymbol{w}(s,t)| &= \left| \int \boldsymbol{v}(s,x)\boldsymbol{w}(x,t)dx \right| \\
&\leq \int |\boldsymbol{v}(s,x)||\boldsymbol{w}(x,t)|dx \\
&\leq \int |\boldsymbol{v}(s,x)|dx\|\boldsymbol{w}(\cdot,t)\|_{L^\infty} \\
&\leq \|\boldsymbol{v}\|_{L^\infty L^1} \|\boldsymbol{w}\|_{L^\infty L^\infty}.
\end{aligned}
$$

□

*Proof of Proposition A.3:* Let us look at the residual integral of $\boldsymbol{v} \star \boldsymbol{w}$. For any $s \in \mathbb{R}$ and $d \geq 0$ we have

$$ 
\begin{aligned}
R_{\boldsymbol{v} \star \boldsymbol{w}}(s, s+d) &= \int_{s+d}^{\infty} \int \boldsymbol{v}(s,x)\boldsymbol{w}(x,t)dxdt \\
&= \int \int_{s+d}^{\infty} \boldsymbol{u}(s,x)\boldsymbol{w}(x,t)dtdx \\
&= \int \boldsymbol{v}(s,x)R_{\boldsymbol{w}}(x, s+d)dx.
\end{aligned}
$$

By splitting the integral at $s + pd$ with $p \in [0, 1]$ we have

$$ 
\begin{aligned}
R_{\boldsymbol{v} \star \boldsymbol{w}}(s, s+d) &= \int_{-\infty}^{s+pd} \boldsymbol{v}(s,x)R_{\boldsymbol{w}}(x, s+d)dx + \int_{s+pd}^{\infty} \boldsymbol{v}(s,x)R_{\boldsymbol{w}}(x, s+d)dx \\
&\leq \int_{-\infty}^{s+pd} \boldsymbol{v}(s,x)R_{\boldsymbol{w}}^{\infty}(s+d-x)dx + \int_{s+pd}^{\infty} \boldsymbol{v}(s,x)\|R_{\boldsymbol{w}}\|_{L^{\infty}L^{\infty}}dx \\
&\leq \int_{-\infty}^{s+pd} \boldsymbol{v}(s,x)dxR_{\boldsymbol{w}}^{\infty}(d-pd) + \int_{s+pd}^{\infty} \boldsymbol{v}(s,x)dx\|\boldsymbol{w}\|_{L^{\infty}L^{1}} \\
&\leq \|\boldsymbol{v}\|_{L^{\infty}L^{1}}R_{\boldsymbol{w}}^{\infty}(qd) + R_{\boldsymbol{v}}(s, s+pd)\|\boldsymbol{w}\|_{L^{\infty}L^{1}}.
\end{aligned}
$$

Taking the $L^\infty$ norm on $s$ on both side leads to the result.

□
*Proof of Proposition A.5:* It is clear that $\psi$ is well defined since

$$ \left\| \sum_{n \ge 1} v^{\star n} \right\|_{L^\infty L^1} \le \sum_{n \ge 1} V^n \le V (\mathrm{Id} - V)^{-1}. $$

Recall that $\delta^0: (s,t) \mapsto \delta_0(t-s)$ Id is the unit for $\star$. Then we have

$$ \psi = v \star (\psi + \delta^0). $$

Since $R_{\delta^0}^\infty(t) = 0$ for $t > 0$, applying Proposition A.3 gives

$$ R_{\psi}^{\infty}(d) \le V R_{\psi}^{\infty}(pd) + R_{v}^{\infty}((1-p)d)(\mathrm{Id}-V)^{-1}, \quad d > 0, 0 < p < 1. $$

Iterating $n$ times leads to

$$ R_{\psi}^{\infty}(d) \le \sum_{k=0}^{n-1} V^k R_{v}^{\infty}((1-p)p^k d)(\mathrm{Id}-V)^{-1} + V^n R_{\psi}^{\infty}(p^n d), \quad d > 0, 0 < p < 1. $$

Since $R_{\psi}^{\infty}(t) \le V(\mathrm{Id} - V)^{-1}$ for any $t \ge 0$ and since $\mathrm{SpR}(V) < 1$ it is clear that

$$ R_{\psi}^{\infty}(d) \le \sum_{k=0}^{\infty} V^k R_{v}^{\infty}((1-p)p^k d)(\mathrm{Id}-V)^{-1}, \quad d > 0, 0 < p < 1. $$

□

*Proof of Proposition A.7:* Let $t \ge 1$. We use Proposition A.5 with $p = \exp(-\delta \log(1/\alpha)/\gamma)$ and cut the sum at $k_0 = \lfloor \varepsilon \log t \rfloor \ge 0$ with

$$ \varepsilon = \frac{\gamma}{(1 + \delta) \log(1/\alpha)} $$

Thus we have

$$ \begin{aligned} R_{\psi}^{\infty}(t) &\le \sum_{k=0}^{k_0} V^k R_{v}^{\infty}((1-p)p^k t)(\mathrm{Id}-V)^{-1} + \sum_{k=k_0+1}^{\infty} V^k R_{v}^{\infty}((1-p)p^k t)(\mathrm{Id}-V)^{-1} \\ &\le (\mathrm{Id}-V)^{-1} R_{v}((1-p)p^{k_0} t)(\mathrm{Id}-V)^{-1} + (\mathrm{Id}-V)^{-1} V^{k_0+2}(\mathrm{Id}-V)^{-1}. \end{aligned} $$

By definition we have

$$ (1-p)p^{k_0}t \ge (1-p)p^{\varepsilon \log t}t = (1-p)t^{\frac{1}{1+\delta}} \ge (1-p)t^{1-\delta}. $$

Thus we have

$$ \begin{aligned} R_{v}^{\infty}((1-p)p^{k_0}t) &\le \frac{A}{[1+(1-p)t^{1-\delta}]^{\gamma}} V \\ &\le \frac{A(1-p)^{-\gamma}}{(1+t)^{\gamma(1-\delta)}} V \end{aligned} $$

where we used the fact that

$$ \frac{1}{1+cx^a} \le \frac{c^{-1}}{(1+x)^a}, \quad x \ge 0, a, c \in (0,1). $$

Indeed, we have

$$ \frac{(1+x)^a}{1+cx^a} = \frac{(1+x)^a}{1+x^a} \frac{1+x^a}{1+cx^a} \le 1 \times c^{-1}. $$
It proves the first result since $V$ commutes with $(\mathrm{Id}-V)^{-1}$. Let us now check the second bound on the norm. Since we have for $j \ge 1$,

$$ \|V^j(\mathrm{Id}-V)^{-1}\|_{\infty} \le \sum_{i \ge j} K \alpha^i = \frac{K \alpha^j}{1-\alpha}, $$

and that

$$ \|(\mathrm{Id}-V)^{-1}\|_{\infty} \le 1 + \sum_{i \ge 1} K \alpha^i \le \frac{K+1}{1-\alpha}, $$

it is clear that

$$ \|R_{\psi}^{\infty}(t)\|_{\infty} \le \frac{K(K+1)\alpha}{(1-\alpha)^2} \times \frac{A(1-p)^{-\gamma}}{(1+t)^{\gamma(1-\delta)}} + \frac{K(K+1)\alpha^{k_0+2}}{(1-\alpha)^2}. $$

Since we have

$$ \alpha^{k_0+2} \le \alpha \alpha^{\varepsilon \log(t)} = \alpha t^{-\gamma/(1+\delta)} \le \alpha t^{-\gamma(1-\delta)} \le \alpha 2^{\gamma(1-\delta)} (1+t)^{-\gamma(1-\delta)}, $$

we conclude that

$$ \|R_{\psi}^{\infty}(t)\|_{\infty} \le K(K+1)\alpha \frac{A(1-p)^{-\gamma} + 2^{\gamma(1-\delta)}}{(1-\alpha)^2(1+t)^{\gamma(1-\delta)}}, \quad t \ge 1. $$

Finally, for $x > 0$ we have $(1 - e^{-x})^{-1} \le 1 + x^{-1}$ thus

$$ (1-p)^{-1} = \left(1 - e^{-\frac{\delta \log(1/\alpha)}{\gamma}}\right)^{-1} \\ \le 1 + \frac{\gamma}{\delta \log(1/\alpha)}. $$

Thus the bound is clear.

□

*Proof of Proposition A.8:* For $0 < \varepsilon < \frac{c(1-\alpha)}{(1-\alpha)+A\alpha}$ consider $v_{\varepsilon}(s,t) = v(s,t)e^{\varepsilon(t-s)}$. We can express $\psi$ as follows,

$$ \psi(s,t) = e^{-\varepsilon(t-s)} \sum_{n \ge 1} v_{\varepsilon}^{*n}(s,t) = e^{-\varepsilon(t-s)} \psi_{\varepsilon}(s,t). $$

Thus we have

$$ R_{\psi}^{\infty}(t) \le e^{-\varepsilon t} R_{\psi_{\varepsilon}}^{\infty}(t). $$

Then, for any $t$ we have $R_{\psi_{\varepsilon}}^{\infty}(t) \le V_{\varepsilon}(\mathrm{Id}-V_{\varepsilon})^{-1}$ where

$$ V_{\varepsilon} = \|v_{\varepsilon}\|_{L^{\infty}L^{1}}. $$

Let $s \in \mathbb{R}$, we have

$$ \begin{aligned} \int v(s,t) e^{\varepsilon(t-s)} dt &= \left[-R_v(s,t) e^{\varepsilon(t-s)}\right]_{t=-\infty}^{t=\infty} + \varepsilon \int R_v(s,t) e^{\varepsilon(t-s)} dt \\ &\le 0 + \varepsilon \int_{-\infty}^{s} R_v(s,t) e^{\varepsilon(t-s)} dt + \varepsilon \int_{s}^{\infty} R_v(s,t) e^{\varepsilon(t-s)} dt \\ &\le \varepsilon \int_{-\infty}^{0} e^{\varepsilon t} dt \cdot V + A\varepsilon \int_{0}^{\infty} e^{-(c-\varepsilon)t} dt \cdot V \\ &= \left(1 + \frac{A\varepsilon}{c-\varepsilon}\right) V. \end{aligned} $$
Thus $V_ε \preceq (1 + \frac{A_ε}{c-ε})V$. The condition on $ε$ ensures that $α(1 + \frac{A_ε}{c-ε}) < 1$ and thus $SpR(V_ε) < 1$. In fact we have $V_ε \in Ge(α(1 + \frac{A_ε}{c-ε}), K)$, from which the bound

$$ \|R_{\psi}^{\infty}(t)\|_{\infty} \le \frac{K\alpha(1 + \frac{A_\epsilon}{c-\epsilon})}{1 - \alpha(1 + \frac{A_\epsilon}{c-\epsilon})}e^{-\epsilon t} $$

is clear, which concludes the proof. □

## Acknowledgements

I would like to thank my PhD advisors, Vincent Rivoirard and Patricia Reynaud-Bouret for their advices and helpful comments.

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