DECOMPOSITIONS OF STOCHASTIC PROCESSES BASED ON IRREDUCIBLE GROUP REPRESENTATIONS

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 [Biometrika, 48 (1961), pp. 109-114]. Relations with Karhunen-Loeve expansions are also discussed, and some further applications and extensions are given-in particular related to Gaussian processes indexed by a torus.