Convergence rates for a branching process in a random environment

TL;DR

This paper investigates the convergence rates of a martingale associated with a supercritical branching process in a random environment, providing explicit conditions and rates for different types of convergence and distributional limits.

Contribution

It offers new explicit convergence rates and norming constants for the martingale in branching processes within random environments, extending understanding of their asymptotic behavior.

Findings

Under a moment condition of order p∈(1,2), W−W_n = o(e^{−na}) a.s.

With a finite exponential moment of W_1, the decay rate of P(|W−W_n| > ε) is supergeometric.

Explicit norming constants a_n(ξ) lead to convergence in law of a_n(ξ)(W−W_n).

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 $EW_1 \log W_1^{\alpha+1} < \infty$ 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.