Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-08T08:38:31.556Z Has data issue: false hasContentIssue false
  • English
  • Français

Limiting behaviour of two-level measure-branching

Part of: Stochastic analysis Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Alison M. Etheridge
Alison M. Etheridge*
Affiliation:
University of Edinburgh
*
* Postal address: Department of Mathematics and Statistics, University of Edinburgh, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

A measure-valued diffusion approximation to a two-level branching structure was introduced in Dawson and Hochberg (1991) where it was shown that conditioned on non-extinction at time t, and appropriately rescaled, the process converges as t → ∞to a non-trivial limiting distribution. Here we discuss a different approach to conditioning on non-extinction (popular in one-level branching) and relate the two limiting distributions.

Keywords

MULTILEVEL BRANCHINGMEASURE-VALUED PROCESSCONDITIONAL LIMIT THEOREM

MSC classification

Primary: 60J57: Multiplicative functionals
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems 60H15: Stochastic partial differential equations
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 4 , December 1993 , pp. 773 - 782
Copyright
Copyright © Applied Probability Trust 1993 

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.)

Footnotes

Research carried out while the author was Neyman Assistant Professor at the University of California, Berkeley.

References

Dawson, D. A. (1977) The critical measure diffusion process. Z. Wahrscheinlichkeitsch. 40, 125–145.Google Scholar
Dawson, D. A. (1991) Measure-valued Markov Processes. Lecture Notes in Mathematics. Springer-Verlag, Berlin.Google Scholar
Dawson, D. A. and Hochberg, K. J. (1991) A multilevel branching model. Adv. Appl. Prob. 23, 701–715.Google Scholar
Etheridge, A. M. (1989) Asymptotic behaviour of some measure-valued diffusions. D. Phil. Thesis, University of Oxford.Google Scholar
Etheridge, A. M. (1990) Asymptotic behaviour of measure-valued critical branching processes. Proc. Amer. Math. Soc. To appear.Google Scholar
Evans, S. N. (1991) Trapping a measure-valued Markov branching process conditioned on non-extinction. Ann. Inst. Henri Poincaré 27, 215–220.Google Scholar
Evans, S. N. (1992) Two representations of a conditioned superprocess. Preprint.Google Scholar
Evans, S. N. and Perkins, E. A. (1990) Measure-valued Markov branching processes conditioned on non-extinction. Israel J. Math. 71, 329–337.Google Scholar
Roelly-Coppoletta, S. and Rouault, A. (1989) Processus de Dawson-Watanabe conditionné par le futur lointain. C.R. Acad. Sci. Paris 309 Sér. I, 867–872.Google Scholar