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

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

2008-11-17

arxiv.org/abs/0811.2583v2

Mathematics

Probability

24 pages; just minor stylistic changes were made in this version

Scientific paper

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.

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