Approximation of a symmetric α-stable Lévy process by a Lévy process with finite moments of all orders

• Autorzy: Z. Michna
• Języki publikacji: EN

Abstrakty

EN

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 $Y_{x}$ 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↓ 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 α-stable random vector is approximated with probability one by a random vector with components having finite second moments.

Kategorie tematyczne

• 60G52: Stable processes
• 60G51: Processes with independent increments; L\'evy processes

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 180
• Numer: 1
• Strony: 1-10

Daty

wydano

2007

Twórcy

autor

Z. Michna

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

Identyfikatory

DOI

10.4064/sm180-1-1