Global limit theorems on the convergence of multidimensional random walks to stable processes

TL;DR

This paper establishes global asymptotic limits for symmetric heavy-tailed random walks on multidimensional integer lattices, showing their convergence to stable processes under certain tail regularity conditions.

Contribution

It provides the first comprehensive global limit theorems for multidimensional heavy-tailed random walks converging to stable processes, emphasizing the importance of tail regularity.

Findings

Transition probabilities exhibit uniform asymptotic behavior

Regularity conditions on tails are crucial for convergence

Results extend classical limit theorems to multidimensional heavy-tailed walks

Abstract

Symmetric heavily tailed random walks on $Z^d, d\geq 1,$ are considered. Under appropriate regularity conditions on the tails of the jump distributions, global (i.e., uniform in $x,t, |x|+t\to\infty,$) asymptotic behavior of the transition probability $p(t,0,x)$ is obtained. The examples indicate that the regularity conditions are essential.