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

Abstract

Symmetric heavily tailed random walks on Zd, d ≥ 1, are considered. Under appropriate regularity conditions on the tails of the jump distributions, global (i.e. uniform in x, t, ∣x∣ + t → ∞,) asymptotic behavior of the transition probability p(t, 0, x) is obtained. The examples indicate that the regularity conditions are essential.