Stable convergence of generalized stochastic integrals and the principle of conditioning: L² theory

Giovanni Peccati (1) , Murad Taqqu (2)

1 LSTA - Laboratoire de Statistique Théorique et Appliquée
2 Department of Mathematics and Statistics [Boston]

Consider generalized adapted stochastic integrals with respect to independently scattered random measures with second moments. We use a decoupling technique, known as the "principle of conditioning" , to study their stable convergence towards mixtures of infinitely divisible distributions. Our results apply, in particular, to multiple integrals with respect to independently scattered and square integrable random measures, as well as to Skorohod integrals on abstract Wiener spaces. As a specific application, we establish a Central Limit Theorem for sequences of double integrals with respect to a general Poisson measure, thus extending the results contained in Nualart and Peccati (2005) and Peccati and Tudor (2004) to a non-Gaussian context.

Mots clés: Generalized stochastic integrals; Independently scattered measures; Decoupling; Principle of conditioning; Resolutions of the identity; Stable convergence; Weak convergence; multiple Poisson integrals; Skorohod integrals

Domaines: Probabilités [math.PR]

Giovanni Peccati, Murad Taqqu. Stable convergence of generalized stochastic integrals and the principle of conditioning: L² theory. 2006. ⟨hal-00023359⟩