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

Discrete approximation of stable white noise - Application to spatial linear filtering