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Stable Central Limit Theorems for Super Ornstein-Uhlenbeck Processes, II

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Abstract

This paper is a continuation of our recent paper (Electron. J. Probab., 24(141), (2019)) and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes (Xt)t≥0 with branching mechanisms of infinite second moments. In the aforementioned paper, we proved stable central limit theorems for Xt(f) for some functions f of polynomial growth in three different regimes. However, we were not able to prove central limit theorems for Xt(f) for all functions f of polynomial growth. In this note, we show that the limiting stable random variables in the three different regimes are independent, and as a consequence, we get stable central limit theorems for Xt(f) for all functions f of polynomial growth.

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Acknowledgements

We thank the referees for the very helpful comments.

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Authors and Affiliations

  1. LMAM School of Mathematical Sciences & Center for Statistical Science, Peking University, Beijing, 100871, P. R. China

    Yan Xia Ren

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

    Ren Ming Song

  3. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P. R. China

    Zhen Yao Sun

  4. The Faculty of Industrial Engineering and Management, Technion, Haifa, 3200003, Israel

    Zhen Yao Sun

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

    Jian Jie Zhao

Authors
  1. Yan Xia Ren
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  2. Ren Ming Song
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  4. Jian Jie Zhao
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Corresponding author

Correspondence to Zhen Yao Sun.

Additional information

Yan Xia Ren is supported in part by NSFC (Grant Nos. 11731009 and 12071011) and the National Key R&D Program of China (Grant No. 2020YFA0712900). Renming Song is supported in part by Simons Foundation (#429343, Renming Song)

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Ren, Y.X., Song, R.M., Sun, Z.Y. et al. Stable Central Limit Theorems for Super Ornstein-Uhlenbeck Processes, II. Acta. Math. Sin.-English Ser. 38, 487–498 (2022). https://doi.org/10.1007/s10114-022-0559-y

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  • DOI: https://doi.org/10.1007/s10114-022-0559-y

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