Skip to main content Accessibility help
×
Home

The Critical Galton-Watson Process Without Further Power Moments

  • S. V. Nagaev (a1) and V. Wachtel (a2)

Abstract

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Z n ; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      The Critical Galton-Watson Process Without Further Power Moments
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      The Critical Galton-Watson Process Without Further Power Moments
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      The Critical Galton-Watson Process Without Further Power Moments
      Available formats
      ×

Copyright

Corresponding author

Postal address: Sobolev Institute for Mathematics, Prospect Akademika Koptjuga 4, 630090 Novosibirsk, Russia.
∗∗ Postal address: Technische Universität München, Zentrum Mathematik, Bereich M5, TU München, 85747 Garching, Germany. Email address: wachtel@ma.tum.de

References

Hide All
[1] Bondarenko, E. M. and Topchii, V. A. (2001). Estimates for expectation of the maximum of a critical Galton–Watson process on a finite interval. Siberian Math. J. 42, 209216.
[2] Borovkov, K. A. (1988). A method for the proof of limit theorems for branching processes. Teor. Verojatn. i Primenen 33, 115123.
[3] Darling, D. A. (1952). The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73, 95107.
[4] Darling, D. A. (1970). The Galton–Watson process with infinite mean. J. Appl. Prob. 7, 455456.
[5] Hudson, I. L. and Seneta, E. (1977). A note on simple branching processes with infinite mean. J. Appl. Prob. 14, 836842.
[6] Schuh, H.-J. and Barbour, A. D. (1977). On the asymptotic behaviour of branching processes with infinite mean. Adv. Appl. Prob. 9, 681723.
[7] Seneta, E (1976). Regularly Varying Functions (Lecture Notes Math. 508). Springer, Berlin.
[8] Slack, R. S. (1968). A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitsth. 9, 139145.
[9] Slack, R. S. (1972). Further notes on branching process with mean one. Z. Wahrscheinlichkeitsth. 25, 3138.
[10] Sze, M. (1976). Markov processes associated with critical Galton–Watson processes with application to extinction probability. Adv. Appl. Prob. 8, 278295.
[11] Zubkov, A. M. (1975). Limit distributions of the distance to the nearest common ancestor. Teor. Verojatn. i Primenen 20, 614623.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed