Limit theorems for a supercritical multi-type branching process with immigration in a random environment

Let $\{Z_n^i = (Z_n^i(r))_{1 \le r \le d}: n \ge 0\}$ be a supercritical $d$-type branching process in an i.i.d. environment $\xi = (\xi_0, \xi_1, \dots)$, starting from a single particle of type $i$. The offspring distribution at generation $n$ depends on the environment $\xi_n$, and we denote by $M_n = (M_n(i,j))_{1 \le i,j \le d}$ the corresponding (random) mean matrix. Recently, Grama et al. (Ann. Appl. Probab. \textbf{33}(2023) 1213-1251) extended the famous Kesten--Stigum theorem to the random environment case with $d>1$. They improved upon previous work by innovatively constructing a new normalized population process $(\tilde{W}^i_n)$. Under several simple assumptions, they proved that $\tilde{W}^i_n$ converges almost surely to a limit $\tilde{W}^i$, and that $\tilde{W}^i$ is non-degenerate if and only if a $\mbb{E}X\log^+ X<\infty$ type condition holds. In this paper, we study the situation where an immigrant vector $Y_n$ joins the population $Z_n^i$ at each generation $n \ge 0$; the distribution of $Y_n$ also depends on the environment $\xi_n$. Following the approach of Grama et al., we construct a normalized process $(W^i_n)$ for the model with immigration, establishing a Kesten--Stigum type theorem that characterizes the non-degeneracy of its almost sure limit. Moreover, we provide complete $L^p$-convergence criteria for $(W^i_n)$, treating separately the cases $1 < p < \infty$ and $0 < p < 1$. As an important byproduct, a sufficient condition for the boundedness of the maximal function $\sup_n \tilde{W}_n^i$ is also obtained. Our results show that, under a mild restriction on the number of immigrants, the inclusion of immigration does not affect the almost sure convergence property of the original normalized process, but it does have an impact on the criterion for $L^p$ convergence.