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 $X$ , $\tilde{U}=\{{\tilde{U}}_{j,k},j,k\in \overline{\mathbb{N}}\}$, 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 ${\tilde{X}}_{\alpha }=({\tilde{X}}_{\alpha ,1},\dots )$ 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{ η_{j}}{( 2\,ϕ_{j})^{1/2} }=1 \quad a.s. \quad \implies \quad \limsup_{n\to \infty}\frac{\widetilde X_{α, j}}{ϕ_{j} }=1\quad a.s.,\label{a.2} \end{equation} for all $\alpha >0$, and sequences $\varphi =\{{\varphi }_j\}$ such that $f_j=o({\varphi }_j)$. The function $\varphi$ 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 Lévy processes on $\mathbf{Z}$.