Shifted small deviations and Chung LIL for symmetric alpha-stable processes

TL;DR

This paper derives exact small deviation probabilities for symmetric alpha-stable Levy processes with shifts, and establishes a Chung Law of the Iterated Logarithm for these processes, characterizing their asymptotic behavior.

Contribution

It provides the first precise small ball probability estimates with constants for shifted symmetric alpha-stable Levy processes and proves a functional Chung LIL for these processes.

Findings

Exact rate of small ball probability decay including constants.

A functional Chung LIL for symmetric alpha-stable Levy processes.

Characterization of the limit set as all continuous functions starting at 0.

Abstract

Consider a symmetric $\alpha$-stable L\'evy process with $\alpha\in (1,2)$. We study shifted small ball probabilities for these processes in the uniform topology, when the shift function is an arbitrary continuous function which starts at 0. We obtain the exact rate of decrease for these probabilities including constants. Using these small ball estimates, we obtain a functional LIL for $\alpha$-stable L\'evy process with attracting functions that are continuous. It occurs that the limit set for the family of renormalized $\alpha$-stable L\'evy processes is equal to the set of all continuous functions on $[0,1]$ which start at 0, under certain choice of normalizing functions.