Convergence Rates for a Supercritical Branching Process in a Random Environment

Chunmao Huang, Quansheng Liu

2014, v.20, №2, 265-286

ABSTRACT

Let $(Z_n)$ be a supercritical branching process in a stationary and ergodic random environment $\xi$. We study the convergence rates of the martingale $W_n = Z_n/ \mathbb{E}[Z_n| \xi]$ to its limit $W$. The following results about the convergence almost sure (a.s.), in law or in probability, are shown. (1) Under a moment condition of order $p\in (1,2)$, $W-W_n = o (e^{-na})$ a.s. for some $a>0$ that we find explicitly; assuming only $\mathbb{E} W_1 \log W_1^{\alpha +1} < \infty$ for some $\alpha >0$, we have $W-W_n = o (n^{-\alpha})$ a.s.; similar conclusions hold for a branching process in a varying environment. (2) Under a second moment condition, there are norming constants $a_n(\xi)$ (that we calculate explicitly) such that $a_n(\xi) (W-W_n)$ converges in law to a non-degenerate distribution. (3) For a branching process in a finite state random environment, if $W_1$ has a finite exponential moment, then so does $W$, and the decay rate of $\mathbb{P}(|W-W_n| > \epsilon)$ is supergeometric.

Keywords: branching process,varying environment,random environment,martingale,convergence rates,convergence in law,exponential moment

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