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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Abstract

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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.

How to cite

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Z. 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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