Convergence in $L^p$ for a supercritical multi-type branching process in a random environment

Consider a $d$-type supercritical branching process $Z_n^{i} =(Z_n^i(1), \cdots, Z_n^i(d)),\,n\geq 0,$ 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 $1\leq i,j\leq d$, $Z_n^{i}(j)/\mathbb{E}_\xi Z_n^{i}(j) \rightarrow W^{i}$ in probability as $n \rightarrow +\infty$, 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 $Z_n^{i}(j)/\mathbb{E}_\xi Z_n^{i}(j)$, for each given $p>1$, 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 $(Z_n^{i})$