TY - JOUR
T1 - Limit theorems for a class of critical superprocesses with stable branching
AU - Ren, Yan Xia
AU - Song, Renming
AU - Sun, Zhenyao
N1 - Funding Information:
The research of Yan-Xia Ren is supported in part by NSFC (Grant Nos. 11671017 and 11731009), and LMEQF (Key Laboratory of Mathematical Economics and Quantitative Finance, Peking University, Ministry of Education, China).The research of Renming Song is supported in part by the Simons Foundation (#429343, Renming Song).
Publisher Copyright:
© 2020 Elsevier B.V.
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 -