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

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Michna Z.

Studia Mathematica

2007

tom 180

nr 1

1-10

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.

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1 Studia Mathematica

1 Michna Z.

1 2007

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