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

Published online by Cambridge University Press:  19 September 2016

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: [email protected]

Abstract

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→∞.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2016 

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