Convergence rate for a class of supercritical superprocesses

TL;DR

This paper investigates the almost sure and $L^p$ convergence rates of a martingale associated with a supercritical superprocess, especially when the process lacks finite variance, providing necessary and sufficient conditions.

Contribution

It offers new insights into the convergence rates of superprocess martingales without requiring finite variance, extending previous results.

Findings

Established necessary and sufficient conditions for convergence rates.

Derived convergence rate results in $L^p$ spaces for $p o 1, 2$.

Analyzed the behavior of the martingale difference $M_t()-M_$ as $t o .

Abstract

Suppose $X=\{X_t, t\ge 0\}$ is a supercritical superprocess. Let $\phi$ be the non-negative eigenfunction of the mean semigroup of $X$ corresponding to the principal eigenvalue $\lambda>0$. Then $M_t(\phi)=e^{-\lambda t}\langle\phi, X_t\rangle, t\geq 0,$ is a non-negative martingale with almost sure limit $M_\infty(\phi)$. In this paper we study the rate at which $M_t(\phi)-M_\infty(\phi)$ converges to $0$ as $t\to \infty$ when the process may not have finite variance. Under some conditions on the mean semigroup, we provide sufficient and necessary conditions for the rate in the almost sure sense. Some results on the convergence rate in $L^p$ with $p\in(1, 2)$ are also obtained.