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The Galton-Watson process revisited: some martingale relationships and applications

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

James D. Lynch
James D. Lynch*
Affiliation:
University of South Carolina
*
∗Postal address: University of South Carolina, Center for Reliability and Quality Sciences, Department of Statistics, University of South Carolina, Columbia, SC 29208, USA. Email address: [email protected]
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Abstract

A martingale is used to study extinction probabilities of the Galton-Watson process using a stopping time argument. This same martingale defines a martingale function in its argument s; consequently, its derivative is also a martingale. The argument s can be classified as regular or irregular and this classification dictates very different behavior of the Galton-Watson process. For example, it is shown that irregularity of a point s is equivalent to the derivative martingale sequence at s being closable, (i.e., it has limit which, when attached to the original sequence, the martingale structure is retained). It is also shown that for irregular points the limit of the derivative is the derivative of the limit, and two different types of norming constants for the asymptotics of the Galton-Watson process are asymptotically equivalent only for irregular points.

Keywords

MartingaleGalton-Watson processsupercritical caseexplosive caseregular and irregular pointsderivativeclosable

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G42: Martingales with discrete parameter
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 92D25: Population dynamics (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 322 - 328
Copyright
Copyright © by the Applied Probability Trust 2000 

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Footnotes

Supported by NSF grant DMS 9503104 and DMS 9877107.

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