Density estimation by orthogonal series in an infinite dimensional space: Application to processes of diffusion type I

Abstract

In this paper, we will be concerned with the estimation of the probability density of a diffusion process with respect to the Wiener measure. Since a diffusion process can be viewed as a random variable taking its values in an infinite dimensional space, this problem can be viewed as a special case of estimating the probability density of a random variable with values in an infinite dimensional space. However, obtaining the absolute continuity of probability measures in infinite dimension is difficult. A density estimator based on a Fourier–Hermite orthogonal series is investigated, which is a generalization of the classical density estimator of a real valued random variable. We establish that our estimator converges in integrated and simple mean square, under suitable conditions. Rates of convergence are also given.