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In this paper we consider a symmetric α-stable Lévy process Z. We use a series representation of Z to condition it on the largest jump. Under this condition, Z can be presented as a sum of two independent processes. One of them is a Lévy process Yx parametrized by x > 0 which has finite moments of all orders. We show that Yx converges to Z uniformly on compact sets with probability one as x ↓ 0. The first term in the cumulant expansion of Yx corresponds to a Brownian motion which implies that Yx can be approximated by Brownian motion when x is large. We also study integrals of a nonrandom function with respect to Yx and derive the covariance function of those integrals. A symmetric α-stable random vector is approximated with probability one by a random vector with components having finite second moments.

Approximation of a symmetric α-stable Lévy process by a Lévy process with finite moments of all orders