Convergence rates for a branching process in a random environment

Abstract

Let $(Z_n)$ be a supercritical branching process in a random environment $\xi$. We study the convergence rates of the martingale $W_n=Z_n/E[Z_n|\xi ]$ to its limit $W$. The following results about the convergence almost sur (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 $E{W}_1\log W_1^{\alpha +1}<\infty$ 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 $P(|W-W_n|>\epsilon )$ is supergeometric.

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