Expansions for Gaussian processes and Parseval frames

TL;DR

This paper establishes a connection between series expansions of Gaussian vectors in Banach spaces and Parseval frames in their RKHS, leading to new optimal expansions for fractional Ornstein-Uhlenbeck processes and Gaussian stationary processes.

Contribution

It introduces a novel link between Gaussian series expansions and Parseval frames, and derives optimal expansions for specific Gaussian processes.

Findings

New optimal expansion for fractional Ornstein-Uhlenbeck processes

Extension of results to Gaussian stationary processes with convex covariance

Establishment of a precise link between Gaussian series and Parseval frames

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. In the end an extension of this result to Gaussian stationary processes with convex covariance function is established.