Asymptotic properties of permanental sequences

Abstract

Let $U=\{U_{j,k},j,k\in \overline {\mathbb N}\}$ be the potential of a transient symmetric Borel right process $X$ with state space $\overline {\mathbb N}$. For any excessive function $f=\{f_{k,k\in \overline {\mathbb N}}\}$ for $X$ , $\widetilde U=\{\widetilde U_{j,k},j,k\in\overline {\mathbb N}\}$, where \begin{equation} \widetilde U_{j,k}= U_{j,k} +f_{ k},\qquad j,k\in\overline {\mathbb N},\label{a.1} \end{equation} is the kernel of an $\alpha$-permanental sequence $\widetilde X_{\alpha}=(\widetilde X_{\alpha, 1} ,\ldots)$ for all $\alpha>0$. The symmetric potential $U$ is also the covariance of a mean zero Gaussian sequence $\eta=\{\eta_{j},j\in \overline {\mathbb N}\}$. Conditions are given on the potentials $U$ and excessive functions $f$ under which, \begin{equation} \limsup_{j\to \infty}\frac{ \eta_{j}}{( 2\,\phi_{j})^{1/2} }=1 \quad a.s. \quad \implies \quad \limsup_{n\to \infty}\frac{\widetilde X_{\alpha, j}}{\phi_{j} }=1\quad a.s.,\label{a.2} \end{equation} for all $\alpha>0$, and sequences $\phi=\{\phi_{j}\}$ such that $f_{j}=o(\phi_{j})$. The function $\phi$ is determined by $U$. Many examples are given in which $U$ is the potential of symmetric birth and death processes with and without emigration, first and higher order Gaussian autoregressive sequences and L\'evy processes on $\mathbf Z$.