Decompositions of stochastic processes based on irreductible group representations

2005/09/23 by Giovanni Peccati, Peccati, Giovanni, Jean-Renaud Pycke +1
Mathematics · #FOS: Mathematics #MCS: 60G15 60G07 60H05 20C15 #Probability (math.PR) #math.PR #msc:20C15 #msc:60G07 #msc:60G15 #msc:60H05

paper · pdf · doi:10.48550/arxiv.math/0509569

27 pages

arxiv created 2005/09/23 · arxiv updated 2009/12/01

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-Loève expansions are discussed, and some applications and extensions are given - in particular related to Gaussian processes indexed by a torus.

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