Illinois Journal of Mathematics

Limit theorems for some critical superprocesses

Yan-Xia Ren, Renming Song, and Rui Zhang

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Abstract

Let $X=\{X_{t},t\ge0;\mathbb{P}_{\mu}\}$ be a critical superprocess starting from a finite measure $\mu$. Under some conditions, we first prove that $\lim_{t\to\infty}t{ \mathbb{P}}_{\mu}(\Vert X_{t}\Vert \ne0)=\nu^{-1}\langle\phi_{0},\mu\rangle$, where $\phi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the infinitesimal generator $L$ of the mean semigroup of $X$, and $\nu$ is a positive constant. Then we show that, for a large class of functions $f$, conditioning on $\Vert X_{t}\Vert \ne0$, $t^{-1}\langle f,X_{t}\rangle$ converges in distribution to $\langle f,\psi_{0}\rangle_{m}W$, where $W$ is an exponential random variable, and $\psi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the dual of $L$. Finally, if $\langle f,\psi_{0}\rangle_{m}=0$, we prove that, conditioning on $\Vert X_{t}\Vert \ne0$, $(t^{-1}\langle\phi_{0},X_{t}\rangle,t^{-1/2}\langle f,X_{t}\rangle )$ converges in distribution to $(W,G(f)\sqrt{W})$, where $G(f)\sim\mathcal{N}(0,\sigma_{f}^{2})$ is a normal random variable, and $W$ and $G(f)$ are independent.

Article information

Source
Illinois J. Math., Volume 59, Number 1 (2015), 235-276.

Dates
Received: 17 August 2015
Revised: 16 November 2015
First available in Project Euclid: 11 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1455203166

Digital Object Identifier
doi:10.1215/ijm/1455203166

Mathematical Reviews number (MathSciNet)
MR3459635

Zentralblatt MATH identifier
1338.60074

Subjects
Primary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Citation

Ren, Yan-Xia; Song, Renming; Zhang, Rui. Limit theorems for some critical superprocesses. Illinois J. Math. 59 (2015), no. 1, 235--276. doi:10.1215/ijm/1455203166. https://projecteuclid.org/euclid.ijm/1455203166


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