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On large deviation rates for sums associated with Galton‒Watson processes

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  19 September 2016

Hui He
Hui He*
Affiliation:
Beijing Normal University
*
* Postal address: Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China. Email address: hehui@bnu.edu.cn
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Abstract

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Given a supercritical Galton‒Watson process {Zn} and a positive sequence {εn}, we study the limiting behaviors of ℙ(SZn/Zn≥εn) with sums Sn of independent and identically distributed random variables Xi and m=𝔼[Z1]. We assume that we are in the Schröder case with 𝔼Z1 log Z1<∞ and X1 is in the domain of attraction of an α-stable law with 0<α<2. As a by-product, when Z1 is subexponentially distributed, we further obtain the convergence rate of Zn+1/Zn to m as n→∞.

Keywords

Galton‒Watson processdomain of attractionstable distributionslowly varying functionlarge deviationLotka‒Nagaev estimatorSchröder constant

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 672 - 690
Copyright
Copyright © Applied Probability Trust 2016 

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