Functional CLT for the range of stable random walks

TL;DR

This paper proves a functional central limit theorem for the capacity and size of the range of certain stable random walks on integer lattices, extending understanding of their asymptotic behavior in high dimensions.

Contribution

It establishes the first functional CLT for the capacity of the range of $eta$-stable random walks and also for the cardinality of the range in high dimensions.

Findings

Functional CLT for the capacity of the range.

Functional CLT for the cardinality of the range.

Results applicable to $eta$-stable random walks in high dimensions.

Abstract

In this note, we establish a functional central limit theorem for the capacity of the range for a class of $\alpha$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d > 5\alpha/2$. Using similar methods, we also prove an analogous result for the cardinality of the range when $d > 3\alpha / 2$.