Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T01:56:24.134Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes Markov 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]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported by NSF grant DMS 9503104 and DMS 9877107.

References

Athreya, K. B., and Ney, P. E. (1972). Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Chung, K. L. (1974). A Course in Probability Theory. Academic Press, San Diego.Google Scholar
Biggins, J. D., and D'Souza, J. C. (1993). The supercritical Galton–Watson process in varying environments–-Seneta–Heyde norming. Stoch. Proc. Appl. 48, 237–249.CrossRefGoogle Scholar
Cohn, H. (1988). Limit behavior for stochastic monotonicity and applications. Adv. Appl. Prob. 20, 331–347.CrossRefGoogle Scholar
Darling, D. A. (1970). The Galton–Watson process with infinite mean. J. Appl. Prob. 7, 455–456.CrossRefGoogle Scholar
D'Souza, J. C., and Biggins, J. D. (1992). The supercritical Galton–Watson process in varying environments. Stoch. Proc. Appl. 42, 39–47.CrossRefGoogle Scholar
D'Souza, J. C. (1994). The rate of growth of the Galton–Watson process in varying environments. Adv. Appl. Prob. 26, 698–714.CrossRefGoogle Scholar
Grey, D. R. (1977). Almost sure convergence in Markov branching processes with infinite mean. J. Appl. Prob. 14, 702–716.CrossRefGoogle Scholar
Heyde, C. C. (1970). Extension of a result of Seneta for the supercritical Galton–Watson process. Ann. Math. Statist. 41, 739–742.CrossRefGoogle Scholar
Neveu, J. (1975). Discrete Parameter Martingales. North-Holland, Amsterdam.Google Scholar
Sankaranarayan, R. (1989). Branching Processes and its Estimation Theory. John Wiley, New York.Google Scholar
Schuh, H. J., and Barbour, A. D. (1977). On the asymptotic behavior of branching processes with infinite mean. Adv. Appl. Prob. 9, 681–723.CrossRefGoogle Scholar
Seneta, E. (1968). On recent theorems concerning the supercritical Galton–Watson Process. Ann. Math. Statist. 41, 2098–2102.CrossRefGoogle Scholar
Seneta, E. (1969). Functional equations and the Galton–Watson process. Adv. Appl. Prob. 1, 1–42.CrossRefGoogle Scholar