Expansions for Gaussian Processes and Parseval Frames

Harald Luschgy,Gilles Pagès

doi:10.1214/ejp.v14-649

Abstract

We derive a precise link between series expansions of Gaussian random vectors in a Banach space and Parseval frames in their reproducing kernel Hilbert space. The results are applied to pathwise continuous Gaussian processes and a new optimal expansion for fractional Ornstein-Uhlenbeck processes is derived.

Highlights

Series expansions is a classical issue in the theory of Gaussian measures
Our motivation for a new look on this issue finds its origin in recent new expansions for fractional Brownian motions
In this article we are interested in series expansions of X of the following type

Summary

Introduction

Series expansions is a classical issue in the theory of Gaussian measures (see [2], [9], [17]). A sequence ( f j)j≥1 in a real separable Hilbert space (H, (·, ·)) is called Parseval frame for H if ( f j, h) f j converges in H and j=1. Finite sequences in H may serve as frames. ( f j) is admissible for X if and only if ( f j) is a Parseval frame for the reproducing kernel Hilbert space of X. We demonstrate that the right notion of a ”basis” in connection with expansions of X is a Parseval frame and not an orthonormal basis for the reproducing kernel Hilbert space of X. It is convenient to use the symbols ∼ and ≈ where an ∼ bn means an/bn → 1 and an ≈ bn means 0 < lim inf an/bn ≤ lim sup an/bn < ∞

The Banach space setting

Continuous Gaussian processes

Expansion of fractional Ornstein-Uhlenbeck processes

Expansion of stationary Gaussian processes with convex covariance function