Limit theorems for a class of critical superprocesses with stable branching

2018/07/08 by Yan-Xia Ren, Renming Song, Ren, Yan-Xia +3 · 1 citation
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1807.02837

openalex publication_date 2018/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a critical superprocess \X;\mathbf 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, \mathbf 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);\mathbf P_μ(⋅|‖X_t‖≠ 0)\, after appropriate rescaling, converges weakly to a positive random variable \mathbf z^(γ₀-1) with Laplace transform E[e^-u\mathbf z^(γ₀-1)]=1-(1+u^-(γ₀-1))^-1/(γ₀-1).

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• Critical non-local spatial branching processes with infinite variance conditioned on survival

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