Convergence rate for a class of supercritical superprocesses

Suppose $X=\{X_t,t\ge 0\}$ is a supercritical superprocess. Let $\varphi$ 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 $M_{\mathrm{\infty }}(\varphi )$. In this paper we study the rate at which $M_t(\varphi )-M_{\mathrm{\infty }}(\varphi )$ converges to $0$ as $t\to \mathrm{\infty }$ 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.