Limit theorems for a stable sausage

Abstract

In this article, we study fluctuations of the volume of a stable sausage defined via a $d$-dimensional rotationally invariant $\alpha$-stable process. As the main results, we establish a functional central limit theorem (in the case when $d/\alpha >3/2$) with a standard one-dimensional Brownian motion in the limit, and Khintchine's and Chung's laws of the iterated logarithm (in the case when $d/\alpha >9/5$).

Due to a problem in the proof, almost sure invariance principle has been removed and the laws of the iterated logarithms are proved for the case when $d/\alpha >9/5.$