Decompositions of stochastic processes based on irreductible group representations

TL;DR

This paper introduces a novel decomposition method for G-indexed stochastic processes using irreducible group representations, providing insights into Gaussian processes and classical identities like Watson's, with applications to processes on tori.

Contribution

It presents a new decomposition framework based on group representation theory for stochastic processes, extending classical identities and connecting to Karhunen-Loève expansions.

Findings

Decomposition explains Watson's identity for Brownian bridges.

Connections established with Karhunen-Loève expansions.

Applications to Gaussian processes on tori.

Abstract

Let G be a topological compact group acting on some space Y. We study a decomposition of Y-indexed stochastic processes, based on the orthogonality relations between the characters of the irreducible representations of G. In the particular case of a Gaussian process with a G-invariant law, such a decomposition gives a very general explanation of a classic identity in law - between quadratic functionals of a Brownian bridge - due to Watson (1961). Several relations with Karhunen-Lo\`{e}ve expansions are discussed, and some applications and extensions are given - in particular related to Gaussian processes indexed by a torus.