关键词: Branching process with immigration, random environment, convergence rates, central limit theorem, convergence in law, convergence in probability

Abstract: We are interested in the convergence rates of the submartingale ${W}_{n} =\frac{Z_{n}}{\Pi_{n}}$ to its limit ${W},$ where $(\Pi_{n})$ is the usually used norming sequence and $(Z_{n})$ is a supercritical branching process with immigration $(Y_{n})$ in a stationary and ergodic environment $\xi$. Under suitable conditions, we establish the following central limit theorems and results about the rates of convergence in probability or in law: (i) $W-W_{n}$ with suitable normalization converges to the normal law $N(0,1)$, and similar results also hold for $W_{n+k}-W_{n}$ for each fixed $k\in \mathbb{N}^{\ast};$ (ii) for a branching process with immigration in a finite state random environment, if $W_{1}$ has a finite exponential moment, then so does $W,$ and the decay rate of $\mathbb{P}(|W-W_{n}|>\varepsilon)$ is supergeometric; (iii) there are normalizing constants $a_{n}(\xi)$ (that we calculate explicitly) such that $a_{n}(\xi)(W-W_{n})$ converges in law to a mixture of the Gaussian law.

Key words: Branching process with immigration, random environment, convergence rates, central limit theorem, convergence in law, convergence in probability