Total offspring of Poisson multitype branching process

Question

A normal branching process $Z_n$ initialized with $Z_0=1$ and offspring generated from $Pois(p),p<1,$ has a total progeny / total off spring distribution

$$X=\sum_{n=0}^\infty Z_n$$ $X\in \mathbb{N}$ almost surely, because $Z_n$ goes extinct eventually when $p<1$. $X$ follows the Borel Distribution. In particular $\mathbb{E}\exp(cX)<\infty$ for small $c$.

Question: does the total offspring $X$ also satisfy $\mathbb{E}\exp(cX)<\infty$ for a multitype branching process with poisson offspring?

Elaboration: The relevant multitype branching process can be described as a Markov chain $Z_n\in \mathbb{N}^m,$ with a $m\times m$ matrix $A$ as parameter. The distribution $Z^{i}_n\vert Z_n=v$ is equal in law to

$\sum_{j=1}^m\sum_{k=1}^{v_k} Pois(a_{ij})$

where all poisson summands are independent.

Again, this process goes extinct when $A$ has spectral radius strictly smaller than 1.

I am interested whether the variable

$$X=\sum_{n=0}^\infty\sum_{j=1}^m Z^{j}_n$$

satisfies $\mathbb{E}\exp(cX)<\infty$ for small $c$.

Answer 1

Yes, with essentially the same proof. In the scalar case, just notice that $e^{cX_n+bZ_n}$ is a supermartingale as long as $p(e^{c+b}-1)\le b$. In the vector case you will need to find positive vectors $c,b\in(0,\infty)^m$ with the property $(e^{c+b}-1)A\le b$ (I use the notation $f(c)=(f(c_1),\dots,f(c_m))$ and compare the vectors entry-wise), which is possible if and only if the spectral radius of the matrix $A$ with non-negative entries is strictly less than $1$.