Convergence rates for a branching process in a random environment

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 c...

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Bibliographic Details
Published inarXiv.org
Main Authors Huang, Chunmao, Liu, Quansheng
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 16.02.2013
Subjects
Branching (mathematics)
Convergence
Decay rate
Markov analysis
Markov processes
Martingales
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Summary: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.
ISSN:2331-8422