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

Yan Xia Ren; Ren Ming Song; Zhen Yao Sun; Jian Jie Zhao

doi:10.1007/s10114-022-0559-y

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

Yan Xia Ren, Ren Ming Song, Zhen Yao Sun^*, Jian Jie Zhao

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

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 (X_t)_t≥0 with branching mechanisms of infinite second moments. In the aforementioned paper, we proved stable central limit theorems for X_t(f) for some functions f of polynomial growth in three different regimes. However, we were not able to prove central limit theorems for X_t(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 X_t(f) for all functions f of polynomial growth.

Original language	English
Pages (from-to)	487-498
Number of pages	12
Journal	Acta Mathematica Sinica, English Series
Volume	38
Issue number	3
DOIs	https://doi.org/10.1007/s10114-022-0559-y
Publication status	Published - Mar 2022
Externally published	Yes

Keywords

60F05
60J68
Ornstein-Uhlenbeck processes
Superprocesses
branching rate regime
central limit theorem
law of large numbers
stable distribution

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

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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author = "Ren, {Yan Xia} and Song, {Ren Ming} and Sun, {Zhen Yao} and Zhao, {Jian Jie}",

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doi = "10.1007/s10114-022-0559-y",

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T1 - Stable Central Limit Theorems for Super Ornstein-Uhlenbeck Processes, II

AU - Ren, Yan Xia

AU - Song, Ren Ming

AU - Sun, Zhen Yao

AU - Zhao, Jian Jie

PY - 2022/3

Y1 - 2022/3

N2 - 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.

AB - 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.

KW - 60F05

KW - 60J68

KW - Ornstein-Uhlenbeck processes

KW - Superprocesses

KW - branching rate regime

KW - central limit theorem

KW - law of large numbers

KW - stable distribution

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

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Keywords

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