Global limit theorems on the convergence of multidimensional random   walks to stable processes
  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. 
A. Agbor, S. Molchanov, B. Vainberg
math.PR