  Motivated by the simulation of stable random fields, we consider the issue of
discrete approximations of independently scattered stable noise. Two approaches
are proposed: grid approximations available when the underlying space is
$\bbR^d$ and shot noise approximations available on more general spaces. Limit
theorems stating the convergence of discrete random noises to stable white
noise are proved. These results are then applied to study moving average
spatial random fields with heavy-tailed innovations and related limit theorems.
A second application deals with discrete approximation for Brownian L\'evy
motion on the sphere or on the euclidean space.
