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

Abstract

Symmetric heavily tailed random walks on $Z^d,d\ge 1,$ 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.

A misprint is corrected in the main Theorem 1.2