Hostname: page-component-7bb8b95d7b-wpx69 Total loading time: 0 Render date: 2024-09-17T06:33:04.212Z Has data issue: false hasContentIssue false
  • English
  • Français

A maximum sequence in a critical multitype branching process

Published online by Cambridge University Press:  14 July 2016

Aurel Spåtaru
Aurel Spåtaru*
Affiliation:
Centre of Mathematical Statistics, Bucharest
*
∗ Postal address: Centre of Mathematical Statistics, Bd. Magheru 22, 70158 Bucharest, Romania.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (Zn) be a p-type positively regular and non-singular critical Galton–Watson process with finite second moments. Associated with the spectral radius 1 of the mean matrix of (Zn) consider the right eigenvector u = (u1, · ··, up) > 0, and set . It is shown that lim inf, lim sup whenever Z0 = i, where .

Keywords

MARTINGALESTOPPING TIMERANDOM WALK
Type
Short Communications
Information
Journal of Applied Probability , Volume 28 , Issue 4 , December 1991 , pp. 893 - 897
Copyright
Copyright © Applied Probability Trust 1991 

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. (1988) On the maximum sequence in a critical branching process. Ann. Prob. 16, 502–507.10.1214/aop/1176991770Google Scholar
[2] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.10.1007/978-3-642-65371-1Google Scholar
[3] Chow, Y. S. and Teicher, H. (1978) Probability Theory, Springer-Verlag, New York.10.1007/978-1-4684-0062-5Google 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, New York.Google Scholar
[6] Kåmmerle, K. and Schuh, H.-J. (1986) The maximum in critical Galton-Watson and birth and death processes. J. Appl. Prob. 23, 601–613.Google Scholar
[7] Pakes, A. G. (1987) Remarks on the maxima of a martingale sequence with application to the simple critical branching process. J. Appl. Prob. 24, 768–772.Google Scholar
[8] Spåtaru, A. (1990) Strengthening the Hsu-Robbins-Erdös theorem. Rev. Roumaine Math. Pures Appl. 35, 463–465.Google Scholar
[9] Weiner, H. (1984) Moments of the maximum in a critical branching process. J. Appl. Prob. 21, 920–923.10.2307/3213708Google Scholar