Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-17T12:53:27.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for some functionals of certain Galton-Watson branching processes

Published online by Cambridge University Press:  01 July 2016

Torgny Lindvall
Torgny Lindvall*
Affiliation:
University of Göteborg, Sweden
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper extends the Feller-Jiřina theorem on the diffusion approximation of Galton-Watson branching processes with reproduction mean close to one, and limit theorems are obtained for several functionals of such processes.

Keywords

GALTON-WATSON BRANCHING PROCESSESLIMITING DIFFUSIONLIMIT THEOREMS FOR FUNCTIONALS
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 2 , June 1974 , pp. 309 - 321
Copyright
Copyright © Applied Probability Trust 1974 

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] Billingsley, P. (1968) Convergence of Probability Measures. John Wiley, New York.Google Scholar
[2] Doob, J. L. (1953) Stochastic Processes. John Wiley, New York.Google Scholar
[3] Fahady, K. S., Quine, M. P. and Vere-Jones, D. (1971) Heavy traffic approximations for the Galton-Watson process. Adv. Appl. Prob. 3, 282–300.Google Scholar
[4] Feller, W. (1951) Diffusion processes in genetics. Proc. Second Berkeley Symp. Math. Statist. Prob. 227–246. University of California Press.Google Scholar
[5] Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. II, 2nd ed. John Wiley, New York.Google Scholar
[6] Gikhman, I. I. and Skorokhod, A. V. (1969) Introduction to the Theory of Random Processes. W. B. Saunders, Philadelphia. (English translation.) Google Scholar
[7] Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
[8] Jiřina, P. (1969) On Feller's branching diffusion processes. Časopis. Pěst. Mat. 94, 84–90.Google Scholar
[9] Kesten, H., Ney, P. and Spitzer, F. (1966) The Galton-Watson process with mean one and finite variance. Theor. Probability Appl. XI, 513–540.Google Scholar
[10] Lamperti, J. (1967) Limiting distributions for branching processes. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 225–241. University of California Press.Google Scholar
[11] Lindvall, T. (1973) Weak convergence of probability measures and random functions in the function space D [0, ∞). J. Appl. Prob. 10, 109–121.Google Scholar
[12] Lindvall, T. (1972) Convergence of critical Galton-Watson branching processes. J. Appl. Prob. 9, 445–450.Google Scholar
[13] Lindvall, T. (1973) Weak convergence in the function space D [0, ∞) and diffusion approximation of certain Galton-Watson branching processes. . Department of Mathematics, University of Göteborg, Sweden.Google Scholar
[14] Pakes, A. G. (1971) Some limit theorems for the total progeny of a branching process. Adv. Appl. Prob. 3, 176–192.Google Scholar