Approximation of a symmetric α-stable Lévy process by a Lévy process with finite moments of all orders
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 parametrized by x > 0 which has finite moments of all orders. We show that converges to Z uniformly on compact sets with probability one as x↓ 0. The first term in the cumulant expansion of corresponds to a Brownian motion which implies that can...