Shifted small deviations and Chung LIL for symmetric alpha-stable processes
Abstract
Description
Consider a symmetric $α$-stable Lévy process with $α\in (1,2)$. 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 $α$-stable Lévy process with attracting functions that are continuous. It occurs that the limit set for the family of renormalized $α$-stable Lévy processes is equal to the set of all continuous functions on $[0,1]$ which start at 0, under certain choice of normalizing functions.
24 pages; just minor stylistic changes were made in this version
24 pages; just minor stylistic changes were made in this version