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

TL;DR

This paper explores the connection between small deviations of symmetric alpha-stable vectors in Banach spaces and the metric entropy of their generating operators, revealing new insights and estimates for various stable processes.

Contribution

It generalizes previous results by linking small deviation probabilities to entropy numbers of random operators, especially highlighting phenomena for diagonal operators.

Findings

Identifies a gap between entropies of original and random operators.

Provides new estimates for small deviation rates of stable processes.

Analyzes entropy properties of diagonal operators in detail.

Abstract

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.