Approximation of a symmetric $\alpha $-stable Lévy process by a Lévy process with finite moments of all orders

Z. Michna¹

¹ Department of Mathematics Wrocław University of Economics 53-345 Wrocław, Poland

Studia Mathematica, Tome 180 (2007) no. 1, pp. 1-10

Résumé

In this paper we consider a symmetric $\alpha $-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 $Y_x$ parametrized by $x>0$ which has finite moments of all orders. We show that $Y_x$ converges to $Z$ uniformly on compact sets with probability one as $x\downarrow 0$. The first term in the cumulant expansion of $Y_x$ corresponds to a Brownian motion which implies that $Y_x$ can be approximated by Brownian motion when $x$ is large. We also study integrals of a non-random function with respect to $Y_x$ and derive the covariance function of those integrals. A symmetric $\alpha $-stable random vector is approximated with probability one by a random vector with components having finite second moments.

Détail

DOI : 10.4064/sm180-1-1

Keywords: paper consider symmetric alpha stable process series representation condition largest jump under condition presented sum independent processes process parametrized which has finite moments orders converges uniformly compact sets probability downarrow first term cumulant expansion corresponds brownian motion which implies approximated brownian motion large study integrals non random function respect derive covariance function those integrals symmetric alpha stable random vector approximated probability random vector components having finite second moments