Spine Decompositions and Limit Theorems for a Class of Critical Superprocesses

Abstract

In this paper we first establish a decomposition theorem for size-biased Poisson random measures. As consequences of this decomposition theorem, we get a spine decomposition theorem and a 2-spine decomposition theorem for some critical superprocesses. Then we use these spine decomposition theorems to give probabilistic proofs of the asymptotic behavior of the survival probability and Yaglom's exponential limit law for critical superprocesses.2010 Mathematics Subject Classification. 60J80, 60F05. Key words and phrases. Critical superprocess, size-biased Poisson random measure, spine decomposition, 2-spine decomposition, asymptotic behavior of the survival probability, Yaglom's exponential limit law, martingale change of measure.

A 2-spine decomposition of the critical Galton-Watson tree and a probabilistic proof of Yaglom’s theorem

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