Norm convergent expansions for Gaussian processes in Banach spaces.

Abstract

Several authors have recently shown that Brownian motion with continuous paths on [ 0 , 1 ] [0,1] can be expanded into a uniformly convergent (a.s.) orthogonal series in terms of a given complete orthonormal system (CONS) in its reproducing kernel Hilbert space (RKHS). In an earlier paper we generalized this result to Gaussian processes with continuous paths. Here we obtain such expansions for a Gaussian random variable taking values in an arbitrary separable Banach space. A related problem is also considered in which starting from a separable Hilbert space H H with a measurable norm | | ⋅ | | 1 || \cdot |{|_1} defined on it, it is shown that the corresponding abstract Wiener process has a | | ⋅ | | 1 ||\cdot |{|_1} -convergent orthogonal expansion in terms of a CONS chosen from H H .