Limit theorems for a class of critical superprocesses with stable branching

Yan Xia Ren; Renming Song; Zhenyao Sun

doi:10.1016/j.spa.2020.01.001

Limit theorems for a class of critical superprocesses with stable branching

Yan Xia Ren, Renming Song, Zhenyao Sun^*

^*Corresponding author for this work

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

5 Citations (Scopus)

Abstract

We consider a critical superprocess {X;P_μ} with general spatial motion and spatially dependent stable branching mechanism with lowest stable index γ₀>1. We first show that, under some conditions, P_μ(|X_t|≠0) converges to 0 as t→∞ and is regularly varying with index (γ₀−1)⁻¹. Then we show that, for a large class of non-negative testing functions f, the distribution of {X_t(f);P_μ(⋅|‖X_t‖≠0)}, after appropriate rescaling, converges weakly to a positive random variable z^(γ₀−1) with Laplace transform E[e^{−uz^(γ₀−1)}]=1−(1+u^{−(γ₀−1)})^{−1∕(γ₀−1)}.

Original language	English
Pages (from-to)	4358-4391
Number of pages	34
Journal	Stochastic Processes and their Applications
Volume	130
Issue number	7
DOIs	https://doi.org/10.1016/j.spa.2020.01.001
Publication status	Published - Jul 2020
Externally published	Yes

Keywords

Critical superprocess
Intrinsic ultracontractivity
Regular variation
Scaling limit
Stable branching

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10.1016/j.spa.2020.01.001

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title = "Limit theorems for a class of critical superprocesses with stable branching",

abstract = "We consider a critical superprocess {X;Pμ} with general spatial motion and spatially dependent stable branching mechanism with lowest stable index γ0>1. We first show that, under some conditions, Pμ(|Xt|≠0) converges to 0 as t→∞ and is regularly varying with index (γ0−1)−1. Then we show that, for a large class of non-negative testing functions f, the distribution of {Xt(f);Pμ(⋅|‖Xt‖≠0)}, after appropriate rescaling, converges weakly to a positive random variable z(γ0−1) with Laplace transform E[e−uz(γ0−1)]=1−(1+u−(γ0−1))−1∕(γ0−1).",

keywords = "Critical superprocess, Intrinsic ultracontractivity, Regular variation, Scaling limit, Stable branching",

author = "Ren, {Yan Xia} and Renming Song and Zhenyao Sun",

note = "Publisher Copyright: {\textcopyright} 2020 Elsevier B.V.",

year = "2020",

month = jul,

doi = "10.1016/j.spa.2020.01.001",

language = "English",

volume = "130",

pages = "4358--4391",

journal = "Stochastic Processes and their Applications",

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TY - JOUR

T1 - Limit theorems for a class of critical superprocesses with stable branching

AU - Ren, Yan Xia

AU - Song, Renming

AU - Sun, Zhenyao

PY - 2020/7

Y1 - 2020/7

N2 - We consider a critical superprocess {X;Pμ} with general spatial motion and spatially dependent stable branching mechanism with lowest stable index γ0>1. We first show that, under some conditions, Pμ(|Xt|≠0) converges to 0 as t→∞ and is regularly varying with index (γ0−1)−1. Then we show that, for a large class of non-negative testing functions f, the distribution of {Xt(f);Pμ(⋅|‖Xt‖≠0)}, after appropriate rescaling, converges weakly to a positive random variable z(γ0−1) with Laplace transform E[e−uz(γ0−1)]=1−(1+u−(γ0−1))−1∕(γ0−1).

AB - We consider a critical superprocess {X;Pμ} with general spatial motion and spatially dependent stable branching mechanism with lowest stable index γ0>1. We first show that, under some conditions, Pμ(|Xt|≠0) converges to 0 as t→∞ and is regularly varying with index (γ0−1)−1. Then we show that, for a large class of non-negative testing functions f, the distribution of {Xt(f);Pμ(⋅|‖Xt‖≠0)}, after appropriate rescaling, converges weakly to a positive random variable z(γ0−1) with Laplace transform E[e−uz(γ0−1)]=1−(1+u−(γ0−1))−1∕(γ0−1).

KW - Critical superprocess

KW - Intrinsic ultracontractivity

KW - Regular variation

KW - Scaling limit

KW - Stable branching

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U2 - 10.1016/j.spa.2020.01.001

DO - 10.1016/j.spa.2020.01.001

M3 - Article

AN - SCOPUS:85077999598

SN - 0304-4149

VL - 130

SP - 4358

EP - 4391

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 7

ER -

Limit theorems for a class of critical superprocesses with stable branching

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Keywords

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