Limit theorems for a stable sausage

In this article, we study fluctuations of the volume of a stable sausage defined via a $d$-dimensional rotationally invariant $\alpha$-stable process with $d>3\alpha /2$, and a closed unit ball. As the main results, we establish a functional central limit theorem with a standard one-dimensional Brownian motion in the limit, and an almost sure invariance principle for the process of the volume of a stable sausage. As a consequence, we obtain Khintchine's and Chung's laws of the iterated logarithm for this process.