Invariance principle for the capacity and the cardinality of the range of stable random walks

TL;DR

This paper establishes an invariance principle for the capacity and range size of certain stable random walks on integer lattices, leading to laws of the iterated logarithm for these processes.

Contribution

It introduces an almost sure invariance principle for capacity and range of stable random walks, a novel result in this context.

Findings

Proves invariance principle for capacity and range of stable random walks.

Derives Khintchine's and Chung's laws of the iterated logarithm for these processes.

Applicable for walks with specific dimension and stability index conditions.

Abstract

We prove an almost sure invariance principle for the capacity and the cardinality of the range of a class of $\alpha$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d/\alpha > 5/2$, and $d/\alpha >3/2$, respectively. As a direct consequence, we conclude Khintchine's and Chung's laws of the iterated logarithm for both processes.