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The maximum in critical Galton–Watson and birth and death processes

Published online by Cambridge University Press:  14 July 2016

K. Kämmerle  and
H.-J. Schuh
K. Kämmerle
Affiliation:
Johannes Gutenberg-Universität Mainz
H.-J. Schuh*
Affiliation:
Johannes Gutenberg-Universität Mainz
*
∗Postal address: Johannes Gutenberg-Universität Mainz, Fachbereich 17 Mathematik, Saarstr. 21, 6500 Mainz, W. Germany.
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Abstract

In this paper recent results by Weiner [10] on Mn:= max{Z0, · ··, Zn} are strengthened and generalized, where (Zn)n is a critical Galton–Watson branching process with finite and positive offspring variance and Z0 ≡ 1. It is shown that Explicit asymptotic bounds are given for with . If (Zn)n has a linear fractional offspring distribution, it can be embedded in a critical birth and death process (Ẑ t)t. Using martingale methods one obtains thereof.

These results generalize to the case Z0 ≡ k.

Keywords

STOPPING TIMESTRONG MARKOV PROPERTYMARTINGALE INEQUALITIESOPTIONAL STOPPING THEOREM
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 3 , September 1986 , pp. 601 - 613
Copyright
Copyright © Applied Probability Trust 1986 

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References

[1] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[2] Chow, Y. S. and Teicher, H. (1978) Probability Theory. Springer-Verlag, New York.Google Scholar
[3] Feller, W. (1967/70) An Introduction to Probability Theory and its Applications, Vols. I and II. Wiley, New York.Google Scholar
[4] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[5] Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
[6] Kesten, H. and Stigum, B. P. (1967) Limit theorems for decomposible multi-dimensional Galton–Watson processes. J. Math. Anal. Appl. 17, 309–338.CrossRefGoogle Scholar
[7] Klebaner, F. C. and Schuh, H.-J. (1982) A connection between the limit and the maximum random variable of a branching process in varying environments. J. Appl. Prob. 19, 681–684.Google Scholar
[8] Neveu, J. (1964) Bases mathématiques du calcul des probabilités. Masson et Cie, Paris.Google Scholar
[9] Urbanik, K. (1956) On a problem concerning birth and death processes (in Russian). Acta Math. Acad. Sci. Hungar. 7, 99–106.Google Scholar
[10] Weiner, H. (1984) Moments of the maximum in a critical branching process. J. Appl. Prob. 21, 920–923.Google Scholar
[11] Zolotarev, V. M. (1954) On a problem in the theory of branching random processes (in Russian). Uspehi Mat. Nauk 9, 147–156.Google Scholar