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 be approximated by Brownian motion when x is large. We also study integrals of a non-random function with respect to 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm180-1-1,
author = {Z. Michna},
title = {Approximation of a symmetric $\alpha$-stable L\'evy process by a L\'evy process with finite moments of all orders},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {1-10},
zbl = {1117.60050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm180-1-1}
}
Z. Michna. Approximation of a symmetric α-stable Lévy process by a Lévy process with finite moments of all orders. Studia Mathematica, Tome 178 (2007) pp. 1-10. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm180-1-1/