Convergence in $L^p$ for a Supercritical Multi-type Branching Process in a Random Environment

Consider a $d$-type supercritical branching process $Z_n^i=(Z_n^i(1),\ldots ,Z_n^i(d))$, $n\geq 0$, in an independent and identically distributed random environment $\xi =(\xi _0,\xi _1,\ldots )$, starting with one initial particle of type $i$. In a previous paper we have established a Kesten–Stigum type theorem for $Z_n^i$, which implies that for any $1\leq i,j\leq d$, $Z_n^i(j)/\mathbb E_\xi Z_n^i(j) \to W^i$ in probability as $n \to +\infty $, where $\mathbb E_\xi Z_n^i(j)$ is the conditional expectation of $Z_n^i(j)$ given the environment $\xi $ and $W^i$ is a non-negative and finite random variable. The goal of this paper is to obtain a necessary and sufficient condition for the convergence in $L^p$ of $Z_n^i(j)/\mathbb E_\xi Z_n^i(j)$, and to prove that the convergence rate is exponential. To this end, we first establish the corresponding results for the fundamental martingale $(W_n^i)$ associated to the branching process $(Z_n^i)$.