  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 $\mathbf Z$.
