A canonical form for Gaussian periodic processes

TL;DR

This paper introduces a representation theorem for Gaussian periodic processes, enabling their construction with arbitrary regularity and expressing them as limits of trigonometric series, with regularity detectable through associated sequence decay.

Contribution

It provides a new representation theorem for Gaussian processes, linking regularity to sequence decay and establishing isometric equivalence to l2.

Findings

Gaussian processes can be constructed with arbitrary regularity.

Regularity of paths is detectable via decay rate of associated l2 sequences.

Representation as limits of trigonometric series is established.

Abstract

This article provides a representation theorem for a set of Gaussian processes; this theorem allows to build Gaussian processes with arbitrary regularity and to write them as limit of random trigonometric series. We show via Karhunen-Love theorem that this set is isometrically equivalent to l2. We then prove that regularity of trajectory path of anyone of such processes can be detected just by looking at decrease rate of l2 sequence associated to him via isometry.