Limit Theorems for Some Critical Superprocesses

TL;DR

This paper proves limit theorems for critical superprocesses, including the rate of non-extinction, distributional limits of scaled functionals conditioned on survival, and a central limit theorem under certain conditions.

Contribution

It establishes new conditional limit theorems and a central limit theorem for critical superprocesses, advancing understanding of their long-term behavior.

Findings

Identified the rate of non-extinction for critical superprocesses

Proved the existence of distributional limits of scaled functionals conditioned on non-extinction

Established a central limit theorem for functionals conditioned on survival

Abstract

In this paper we establish some conditional limit theorems for some critical superprocesses $X=\{X_t, t\ge 0\}$. First we identify the rate of non-extinction. Then we show that, for a large class of functions $f$, conditioned on non-extinction at time $t$, the limit, as $t\to\infty$, of $t^{-1}\langle f, X_t\rangle$ exists in distribution and we identify this limit. Finally, we also establish, under some conditions, a central limit theorem for $\langle f, X_t\rangle$ conditioned on non-extinction at time $t$.