Approximation of a symmetric $\alpha $-stable Lévy process by a Lévy process with finite moments of all orders
Tom 180 / 2007
Streszczenie
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.
