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L Log L Criterion for a Class of Superdiffusions

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Rong-Li Liu ,
Yan-Xia Ren  and
Renming Song
Rong-Li Liu*
Affiliation:
Perking University
Yan-Xia Ren*
Affiliation:
Perking University
Renming Song*
Affiliation:
University of Illinois
*
∗Postal address: LMAM School of Mathematical Sciences, Perking University, Beijing, 100871, P. R. China.
∗Postal address: LMAM School of Mathematical Sciences, Perking University, Beijing, 100871, P. R. China.
∗∗∗∗Postal address: Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. Email address: [email protected]
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Abstract

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In Lyons, Pemantle and Peres (1995), a martingale change of measure method was developed in order to give an alternative proof of the Kesten–Stigum L log L theorem for single-type branching processes. Later, this method was extended to prove the L log L theorem for multiple- and general multiple-type branching processes in Biggins and Kyprianou (2004), Kurtz et al. (1997), and Lyons (1997). In this paper we extend this method to a class of superdiffusions and establish a Kesten–Stigum L log L type theorem for superdiffusions. One of our main tools is a spine decomposition of superdiffusions, which is a modification of the one in Englander and Kyprianou (2004).

Keywords

DiffusionssuperdiffusionsPoisson point processKesten–Stigum theoremmartingalemartingale change of measure

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F15: Strong theorems 60J25: Continuous-time Markov processes on general state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 479 - 496
Copyright
Copyright © Applied Probability Trust 2009 

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