Statistical estimation and limit theorems for Gaussian fields using the Malliavin calculus

Overview & Synopsis

In this thesis we apply the Malliavin calculus to statistical estimation of parameters of stochastic processes and to derive limit theorems for the weighted quadratic variations of one or two-parameter fractional processes and to multidimensional normal approximation of probability measures. In Chapter 1 we construct Stein type estimators for the drift of Gaussian processes and for the intensity of Poisson processes. In Chapter 2, we compute the Bayesian estimator of the input of a Poisson channel then extended to normal martingales with chaotic representation property channels. In Chapter 3 we derive central limit theorems for the weighted quadratic variations of the standard Brownian sheet (applied then to the obtaining of an asymptotically normal estimator of the quadratic variation of some two-parameter diffusion processes) and of some fractional Brownian sheets. Then in this chapter we establish a central limit theorem for the weighted quadratic variations of the fractional Brownian motion with Hurst index $H=1/4$ leading to the study of the asymptotic behavior of the Riemann sums with alternating signs associated to the fractional brownian motion with Hurst index $H=1/4$. Finally in Chapter 4 we apply Stein's method and the Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation of functionals of gaussian fields. In particular we provide an application to a functional version of the Breuer-Major TCL for fields subordinated to a fractional Brownian motion.