We characterize the asymptotic independence between blocks consisting of multiple Wiener-Itô integrals. As a consequence of this characterization, we derive the celebrated fourth moment theorem of Nualart and Peccati, its multidimensional extension, and other related results on the multivariate convergence of multiple Wiener-Itô integrals, that involve Gaussian and non Gaussian limits. We give ...

A well-known problem in Malliavin calculus concerns the relation between the determinant of the Malliavin matrix of a random vector and the determinant of its covariance matrix. We give an explicit relation between these two determinants for couples of random vectors of multiple integrals. In particular, if the multiple integrals are of the same order and this order is at most 4, we prove that ...

The convergence in variation of the laws of multiple Wiener–Itô integrals with respect to their kernel has been studied by Davydov and Martynova in [1987. Limit behavior of multiple stochastic integral. Statistics and Control of Random Process (Preila, 1987), Nauka, Moscow, pp. 55–57 (in Russian)]. Here, we generalize this convergence for the joint laws of multiple Wiener–Itô integrals. In this...

In this paper we present a characterization of those Wiener functionals that are the likelihood ratio for a 'signal plus independent noise' model. The characterization is expressed in terms of the representation of such functionals in a series of multiple Wiener integrals. Let {ys, 0 <s =z r} be a random process with measurable sample functions satisfying ] y, ] G K as. Let { Ws, 0 <s < T} be a...

This expository paper is a companion of the four one-hour tutorial lectures given in the occasion of the special month Progress in Stein’s Method, held at the University of Singapore in January 2009. We will explain how one can combine Stein’s method with Malliavin calculus, in order to obtain explicit bounds in the normal and Gamma approximation of functionals of in…nite-dimensional Gaussian …...

For a one-parameter process of the form X, = X0 + & (b, d W, + & & ds, where W is a Wiener process and I+ d W is a stochastic integral, a twice continuously differentiable function f(X,) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. In this paper we present a generalization for the stochastic integrals associated with a tw...

We give a summary on the geometry of iterated Stratonovich integrals. For this exposition, we always have the connection to stochastic Taylor expansion in mind. In particular, we believe that “cubature on Wiener space” is best understood in the setting presented in this text. Besides cubature on Wiener space, we also give a second application regarding the heat kernel on nilpotent free Lie groups.