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Existence and positivity of the limit in processes with a branching structure

Part of: Stochastic processes Limit theorems Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

M. P. Quine  and
W. Szczotka
M. P. Quine*
Affiliation:
University of Sydney
W. Szczotka*
Affiliation:
University of Wrocław
*
∗Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia.
∗∗Postal address: Mathematical Institute, University of Wrocław, Pl. Grunwaldski 2/4, 50–384 Wrocław, Poland.
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Abstract

We define a stochastic process {Xn} based on partial sums of a sequence of integer-valued random variables (K0,K1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K1,K2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {cn} with cn+1/cn → a as n → ∞ such that Xn/cn converges almost surely to a finite random variable which is positive on the event {Xn ↛ 0}. The result is extended to the case of exchangeable summands.

Keywords

Branchingurn modelmartingaleexchangeable

MSC classification

Primary: 60F15: Strong theorems 60G42: Martingales with discrete parameter 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K99: None of the above, but in this section
Secondary: 60G50: Sums of independent random variables; random walks 60G09: Exchangeability
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 132 - 138
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

W. S. partly supported by an Australian Research Council Institutional Grant and by grant KBN, no. 2 PO3A 056 09.

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