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Markov processes associated with critical Galton-Watson processes with application to extinction probabilities

Published online by Cambridge University Press:  01 July 2016

Michael Sze
Michael Sze*
Affiliation:
Ohio State University
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Abstract

As an alternative to the embedding technique of T. E. Harris, S. Karlin and J. McGregor, we show that given a critical Galton–Watson process satisfying some mild assumptions, we can always construct a continuous-time Markov branching process having the same asymptotic behaviour as the given process. Thus, via the associated continuous process, additional information about the original process is obtained. We apply this technique to the study of extinction probabilities of a critical Galton–Watson process, and provide estimates for the extinction probabilities by regularly varying functions.

Keywords

CRITICAL GALTON–WATSON PROCESSMARKOV BRANCHING PROCESSASYMPTOTICALLY EQUIVALENT PROCESSESEXTINCTION TIMEEXTINCTION PROBABILITIESREGULARLY VARYING FUNCTIONS

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 2 , June 1976 , pp. 278 - 295
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer, Berlin.Google Scholar
[2] Bekessy, A. (1957) Eine Verallagemeinerung der Laplaceschen Methode. Publ. Math. Inst. Hung. Acad. Sci. 2, 105–120.Google Scholar
[3] Bojanic, R. and Karamata, J. (1963) On slowly varying functions and asymptotic relations. Mathematics Research Centre Technical Summary Report 432. University of Wisconsin, Madison.Google Scholar
[4] Bojanic, R. and Seneta, E. (1971) Slowly varying functions and asymptotic relations. J. Math. Anal. Appl. 34, 303–315.Google Scholar
[5] Bojanic, R. and Seneta, E. (1973) A unified theory of regularly varying sequences. Math. Z. 134, 91–106.Google Scholar
[6] de Bruijn, N. G. (1959) Pairs of slowly oscillating functions occurring in asymptotic problems concerning the Laplace transform. Nieuw Arch. Wisk. 7, 20–26.Google Scholar
[7] Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
[8] Karamata, J. (1930) Sur un mode de croissance regulière des fonctions. Mathematica (Cluj) 4, 38–53.Google Scholar
[9] Karamata, J. (1933) Sur un mode de croissance regulière, Théorèmes fondamentaux. Bull. Soc. Math. France. 61, 55–62.CrossRefGoogle Scholar
[10] Karlin, S. and McGregor, J. (1968) Embeddability of discrete time simple branching processes into continuous time branching processes. Trans. Amer. Math. Soc. 132, 115–136.Google Scholar
[11] Karlin, S. and McGregor, J. (1968) Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Amer. Math. Soc. 132, 137–145.CrossRefGoogle Scholar
[12] Lamperti, J. (1958) An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc. 88, 380–387.Google Scholar
[13] Seneta, E. (1969) Functional equations and the Galton–Watson process. Adv. Appl. Prob. 1, 1–42.Google Scholar
[14] Galambos, J. and Seneta, E. (1973) Regularly varying sequences. Proc. Amer. Math. Soc. 41, 110–116.Google Scholar
[15] Slack, R. S. (1968) A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitsth. 9, 139–145.Google Scholar
[16] Slack, R. S. (1973) Further notes on branching processes with mean 1. Z. Wahrscheinlichkeitsth. 25, 31–38.Google Scholar