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

Studia Mathematica (2007)

- Volume: 180, Issue: 1, page 1-10
- ISSN: 0039-3223

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topZ. Michna. "Approximation of a symmetric α-stable Lévy process by a Lévy process with finite moments of all orders." Studia Mathematica 180.1 (2007): 1-10. <http://eudml.org/doc/284469>.

@article{Z2007,

abstract = {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.},

author = {Z. Michna},

journal = {Studia Mathematica},

keywords = {symmetric -stable Lévy process; integral with respect to a symmetric -stable Lévy process; Lévy-Itô integral representation; series representation},

language = {eng},

number = {1},

pages = {1-10},

title = {Approximation of a symmetric α-stable Lévy process by a Lévy process with finite moments of all orders},

url = {http://eudml.org/doc/284469},

volume = {180},

year = {2007},

}

TY - JOUR

AU - Z. Michna

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

JO - Studia Mathematica

PY - 2007

VL - 180

IS - 1

SP - 1

EP - 10

AB - 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.

LA - eng

KW - symmetric -stable Lévy process; integral with respect to a symmetric -stable Lévy process; Lévy-Itô integral representation; series representation

UR - http://eudml.org/doc/284469

ER -

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