Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-07-07T08:25:39.510Z Has data issue: false hasContentIssue false
  • English
  • Français

Limiting diffusions for population-size dependent branching processes

Published online by Cambridge University Press:  14 July 2016

Carla Lipow
Carla Lipow*
Affiliation:
University of Pittsburgh
Get access Rights & Permissions [Opens in a new window]

Abstract

A classical theorem on the convergence to a diffusion of a sequence of branching processes will be proved in the case where the branching processes are modified to allow dependence on population size of the generating function governing reproduction. Some recent semigroup convergence theorems due to Kurtz (1969), (1975) provide the main tools.

Keywords

BRANCHING PROCESSPOPULATION-SIZE DEPENDENCEDIFFUSIONSEMIGROUP CONVERGENCEWEAK CONVERGENCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 1 , March 1977 , pp. 14 - 24
Copyright
Copyright © Applied Probability Trust 1977 

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] Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass.Google Scholar
[2] Feller, W. (1951) Diffusion processes in genetics. Proc. 2nd Berkeley Symp. Math. Statist. Prob., 227–246.Google Scholar
[3] Feller, W. (1952) The parabolic differential equations and the associated semigroups of transformations. Ann. Math. 55, 468–519.CrossRefGoogle Scholar
[4] Feller, W. (1954) Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77, 1–31.CrossRefGoogle Scholar
[5] Jagers, P. (1971) Diffusion aproximations to branching processes. Ann. Math. Statist. 42, 2074–2078.CrossRefGoogle Scholar
[6] Jirina, M. (1969) On Feller's branching diffusion processes. Casopis. Pest, Mat. 94, 84–90.CrossRefGoogle Scholar
[7] Kurtz, T. G. (1969) Extensions of Trotter's operator semigroup approximation theorems. J. Funct. Anal. 3, 111–132.CrossRefGoogle Scholar
[8] Kurtz, T. G. (1975) Semigroups of conditioned shifts and approximation of Markov processes. Ann. Prob. 3, 618–642.CrossRefGoogle Scholar