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Spine Decompositions and Limit Theorems for a Class of Critical Superprocesses

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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.

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Acknowledgements

We thank the two referees for very helpful comments on the first version of this paper.

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Author notes

  1. Zhenyao Sun

    Present address: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, P.R. China

    Yan-Xia Ren & Zhenyao Sun

  2. Center for Statistical Science, Peking University, Beijing, 100871, P.R. China

    Yan-Xia Ren

  3. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Renming Song

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  1. Yan-Xia Ren
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Correspondence to Zhenyao Sun.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of Yan-Xia Ren is supported in part by NSFC (Grant Nos. 11671017 and 11731009), and LMEQF.

The research of Renming Song is supported in part by the Simons Foundation (#429343, Renming Song).

Zhenyao Sun is supported by the China Scholarship Council.

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Ren, YX., Song, R. & Sun, Z. Spine Decompositions and Limit Theorems for a Class of Critical Superprocesses. Acta Appl Math 165, 91–131 (2020). https://doi.org/10.1007/s10440-019-00243-7

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