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The maximal J-regular part of a q-variate weakly stationary process

Klotz L.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 1

155--165

Let x be a q-variate (weakly) stationary process over a locally compact Abelian group G, and J a family of subsets of G invariant under translation. We show that the set of all regular non-negative Hermitian matrix-valued measures M not exceeding the (non-stochastic) spectral measure of x and such that the Hilbert space L2(M) is J-regular contains a unique maximal element. Moreover, this maximal element coincides with the spectral measure of the J-regular part of the Wold decomposition of x.