## The Annals of Probability

- Ann. Probab.
- Volume 11, Number 1 (1983), 58-77.

### The Central Limit Theorem for Stochastic Integrals with Respect to Levy Processes

Evarist Gine and Michel B. Marcus

#### 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 $f$.

#### Article information

**Source**

Ann. Probab., Volume 11, Number 1 (1983), 58-77.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993660

**Digital Object Identifier**

doi:10.1214/aop/1176993660

**Mathematical Reviews number (MathSciNet)**

MR682801

**Zentralblatt MATH identifier**

0504.60011

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Secondary: 60H05: Stochastic integrals 60F17: Functional limit theorems; invariance principles

**Keywords**

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

#### Citation

Gine, Evarist; Marcus, Michel B. The Central Limit Theorem for Stochastic Integrals with Respect to Levy Processes. Ann. Probab. 11 (1983), no. 1, 58--77. doi:10.1214/aop/1176993660. https://projecteuclid.org/euclid.aop/1176993660