  We investigate the relation between the small deviation problem for a
symmetric $\alpha$-stable random vector in a Banach space and the metric
entropy properties of the operator generating it. This generalizes former
results due to Li and Linde and to Aurzada. It is shown that this problem is
related to the study of the entropy numbers of a certain random operator. In
some cases, an interesting gap appears between the entropy of the original
operator and that of the random operator generated by it. This phenomenon is
studied thoroughly for diagonal operators. Basic ingredients here are
techniques related to random partitions of the integers. The main result
concerning metric entropy and small deviations allows us to determine or
provide new estimates for the small deviation rate for several symmetric
$\alpha$-stable random processes, including unbounded Riemann--Liouville
processes, weighted Riemann--Liouville processes and the ($d$-dimensional)
$\alpha$-stable sheet.
