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We investigate the relation between the small deviation problem for asymmetric $\alpha$-stable random vector in a Banach space and the metricentropy properties of the operator generating it. This generalizes formerresults due to Li and Linde and to Aurzada. It is shown that this problem isrelated to the study of the entropy numbers of a certain random operator. Insome cases, an interesting gap appears between the entropy of the originaloperator and that of the random operator generated by it. This phenomenon isstudied thoroughly for diagonal operators. Basic ingredients here aretechniques related to random partitions of the integers. The main resultconcerning metric entropy and small deviations allows us to determine orprovide new estimates for the small deviation rate for several symmetric$\alpha$-stable random processes, including unbounded Riemann--Liouvilleprocesses, weighted Riemann--Liouville processes and the ($d$-dimensional)$\alpha$-stable sheet.

Small deviations of stable processes and entropy of the associated random operators