## Electronic Journal of Probability

### Stable convergence of generalized $L^2$ stochastic integrals and the principle of conditioning

#### Abstract

We consider generalized adapted stochastic integrals with respect to independently scattered random measures with second moments, and use a decoupling technique, formulated as a «principle of conditioning», to study their stable convergence towards mixtures of infinitely divisible distributions. The goal of this paper is to develop the theory. Our results apply, in particular, to Skorohod integrals on abstract Wiener spaces, and to multiple integrals with respect to independently scattered and finite variance random measures. The first application is discussed in some detail in the final sectionof the present work, and further extended in a companion paper (Peccati and Taqqu (2006b)). Applications to the stable convergence (in particular, central limit theorems) of multiple Wiener-Itô integrals with respect to independently scattered (and not necessarily Gaussian) random measures are developed in Peccati and Taqqu (2006a, 2007). The present work concludes with an example involving quadratic Brownian functionals.

#### Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 15, 447-480.

Dates
Accepted: 13 April 2007
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464818486

Digital Object Identifier
doi:10.1214/EJP.v12-404

Mathematical Reviews number (MathSciNet)
MR2299924

Zentralblatt MATH identifier
1139.60024

Subjects
Primary: 60G60: Random fields
Secondary: 60G57: Random measures 60F05: Central limit and other weak theorems

Rights

#### Citation

Giovanni, Peccati; Taqqu, Murad. Stable convergence of generalized $L^2$ stochastic integrals and the principle of conditioning. Electron. J. Probab. 12 (2007), paper no. 15, 447--480. doi:10.1214/EJP.v12-404. https://projecteuclid.org/euclid.ejp/1464818486

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