Open Access
August, 1994 Large Deviation Rates for Branching Processes--I. Single Type Case
K. B. Athreya
Ann. Appl. Probab. 4(3): 779-790 (August, 1994). DOI: 10.1214/aoap/1177004971

Abstract

Let {Zn}0 be a Galton-Watson branching process with offspring distribution {pj}0. We assume throughout that p0=0,pj1 for any j1 and 1<m=Σjpj<. Let Wn=Znmm and W=limnWn. In this paper we study the rates of convergence to zero as n of P(|Zn+1Znm|>ε),P(|WnW|>ε), P(|Zn+1Znm∣>ε|Wa) for ε>0 and a>0 under various moment conditions on {pj}. It is shown that the rate for the first one is geometric if p1>0 and supergeometric if p1=0, while the rates for the other two are always supergeometric under a finite moment generating function hypothesis.

Citation

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K. B. Athreya. "Large Deviation Rates for Branching Processes--I. Single Type Case." Ann. Appl. Probab. 4 (3) 779 - 790, August, 1994. https://doi.org/10.1214/aoap/1177004971

Information

Published: August, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0806.60068
MathSciNet: MR1284985
Digital Object Identifier: 10.1214/aoap/1177004971

Subjects:
Primary: 60J80
Secondary: 60F10

Keywords: branching processes , large deviation

Rights: Copyright © 1994 Institute of Mathematical Statistics

Vol.4 • No. 3 • August, 1994
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