CONTRIBUTIONS TO THE STATISTICS OF RANDOM PROCESSES USING MALLIAVIN CALCULUS

In this dissertation we present several applications of Malliavin calculus, both to the statistical analysis of continuous time stochastic processes and to limit theorems for non-linear functionals of Gaussian Fields. Malliavin calculus extends techniques of classical calculus of variations from deterministic functions to random variables. In Malliavin calculus, the so called Malliavin derivative and its adjoint, the divergence operator, are combined with the theory of Hilbert spaces. Just as classical calculus, this theory has proved to be a powerful tool and its applications vary from the existence of densities, to the construction of estimators and the study of weak convergence of sequences of random variables and random vectors, with a special focus on normal approximations.

The first part of the present document is essentially a generalization of a result of Privault and Réveillac (2008), which extends a seminal paper of Stein (1956). Stein has shown that, under certain conditions, there are biased estimators which perform better than the standard estimator for the mean of a multivariate normal vector. It has been shown by Privault and Réveillac that a similar statement holds for Gaussian processes and we shall present a generalization of their work to continuous time models, where the noise is either a chaotic Brownian martingale or a non-martingale noise living in the second Wiener chaos. This first part of the work corresponds to the paper "Drift estimation with non-gaussian noise using Malliavin Calculus" (2015) which has been published by the Electronic Journal of Statistics.

In the second part of the work we give necessary and sufficient criteria for the convergence of sequences of random variables, living in a fixed sum of Wiener chaoses, to a limit which lives in the sum of the first two Wiener chaoses. Our results extend the important findings of Nualart and Peccati (2005), the so-called Fourth Moment Theorem, and a recent finding of Azmoodeh, Peccati and Poly (2014). Our criteria make use of the so-called Gamma-operators which are derived from scalar products of Malliavin derivatives and the infinitesimal generator of the Ornstein-Uhlenbeck semi-group, see for instance Azmoodeh, Peccati and Poly (2014). This part corresponds to the paper "Weak convergence on Wiener space: targeting the first two chaoses" (2017) which has been submitted to the Latin American Journal of Probability and Mathematical Statistics (ALEA).

In the last part of the present work we consider a sequence living in a fixed Wiener chaos and converging in law to a normal variable. A second sequence is supposed to converge in law to a target variable which is the sum of a linear combination of independent chi-square distributed random variables and an independent normal variable. We derive conditions under which the sequence of random vectors, formed by both sequences of random variables, converges in law. We use again Gamma-operators and cumulants to derive necessary and sufficient conditions which can be seen as generalization of results of Peccati and Tudor (2005) for Gaussian limits in the case of sequences of random vectors which converge componentwise. We apply methods developed by Nourdin and Peccati (2009) to examine the rate of convergence of a sequence of double Wiener integrals towards a normal variable.