Cumulants on the Wiener Space

Ivan Nourdin (1) , Giovanni Peccati (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 MODAL'X - Modélisation aléatoire de Paris X

We combine infinite-dimensional integration by parts procedures with a recursive relation on moments (reminiscent of a formula by Barbour (1986)), and deduce explicit expressions for cumulants of functionals of a general Gaussian field. These findings yield a compact formula for cumulants on a fixed Wiener chaos, virtually replacing the usual ``graph/diagram computations'' adopted in most of the probabilistic literature.

Mots clés: Ornstein-Uhlenbeck Semigroup; Malliavin calculus; Gaussian Processes; Cumulants; Diagram Formulae; Ornstein-Uhlenbeck Semigroup

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]

Ivan Nourdin, Giovanni Peccati. Cumulants on the Wiener Space. Journal of Functional Analysis, 2010, 258, pp.3775-3791. ⟨hal-00423831⟩