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February, 1983 The Central Limit Theorem for Stochastic Integrals with Respect to Levy Processes
Evarist Gine, Michel B. Marcus
Ann. Probab. 11(1): 58-77 (February, 1983). DOI: 10.1214/aop/1176993660

Abstract

Let $M$ be a symmetric independently scattered random measure on $\lbrack 0, 1\rbrack$ with control measure $m$ which is uniformly in the domain of normal attraction of a stable measure of index $p \in (0, 2\rbrack$. Let $f$ be a non-anticipating process with respect to $X(t) = M\lbrack 0, t\rbrack$ if $m$ is continuous, and a previsible process in general, satisfying $\int^1_0 E|f|^p dm < \infty$. Then the stochastic integral $\int^t_0 f dM$ can be defined as a process in $D\lbrack 0, 1\rbrack$ and is in the domain of normal attraction of a stable process of order $p$ in $D\lbrack 0, 1\rbrack$ in the sense of of weak convergence of probability measures. If $M$ is Gaussian and continuous in probability then the central limit theorem holds in $C\lbrack 0, 1\rbrack$; in particular, Ito and diffusion processes satisfy the CLT. Our main tool is an upper bound for the weak $L^p$ norm of $\sup_{0 \leq t \leq 1} |\int^t_0 f dM|$ in terms of the $L^p(P \times m)$ norm of $f$.

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Evarist Gine. Michel B. Marcus. "The Central Limit Theorem for Stochastic Integrals with Respect to Levy Processes." Ann. Probab. 11 (1) 58 - 77, February, 1983. https://doi.org/10.1214/aop/1176993660

Information

Published: February, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0504.60011
MathSciNet: MR682801
Digital Object Identifier: 10.1214/aop/1176993660

Subjects:
Primary: 60B12
Secondary: 60F17 , 60H05

Keywords: Domains of attraction in $D\lbrack 0, 1 \rbrack$ and $C\lbrack 0, 1 \rbrack$ , functional central limit theorems , Maximal inequalities , stochastic integrals

Rights: Copyright © 1983 Institute of Mathematical Statistics

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Vol.11 • No. 1 • February, 1983
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