GEOMETRIC STABLE LAWS THROUGH SERIES REPRESENTATIONS

Abstract. Let (Xi) be a sequence of i.i.d. random variables, and let N be a geometric random variable independent of (Xi). Geometric stable distributions are weak limits of (normalized) geometric compounds, SN = X1 + · · · + XN, when the mean of N converges to infinity. By an appro-priate representation of the individual summands in SN we obtain series representation of the limiting geometric stable distribution. In addition, we study the asymptotic behavior of the partial sum process SN (t) = [Nt]∑ i=1 Xi, and derive series representations of the limiting geometric stable process and the corresponding stochastic integral. We also obtain strong invariance principles for stable and geometric stable laws. 1. Introduction. An