Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T16:45:20.396Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

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.)

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