Identities in law between quadratic functionals of bivariate Gaussian processes, through Fubini theorems and symmetric projections

TL;DR

This paper introduces three new identities in law for quadratic functionals of conditioned bivariate Gaussian processes, generalizing classical results and employing advanced stochastic Fubini theorems.

Contribution

It provides novel identities that extend known results for Gaussian process functionals using Fubini theorems and symmetric projections.

Findings

Three new identities in law for quadratic functionals

A two-parameter generalization of Watson's identity

Application of stochastic Fubini theorem to Gaussian measures

Abstract

We present three new identities in law for quadratic functionals of conditioned bivariate Gaussian processes. In particular, our results provide a two-parameter generalization of a celebrated identity in law, involving the path variance of a Brownian bridge, due to Watson (1961). The proof is based on ideas from a recent note by J. R. Pycke (2005) and on the stochastic Fubini theorem for general Gaussian measures proved in Deheuvels et al. (2004).