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Jh-Singularity and Jh-Regularity of Multivariate Stationary Processes Over LCA Groups

Klotz Lutz, Medina Juan Miguel

Probability and Mathematical Statistics

2021

Vol. 41, Fasc. 1

173--192

Let G be an LCA group, Γ its dual group, and H a closed subgroup of G such that its annihilator Λ is countable. Let M denote a regular positive semidefinite matrix-valued Borel measure on Γ and L2(M) the corresponding Hilbert space of matrix-valued functions square-integrable with respect to M. For g∈G, let Zg be the closure in L2(M) of all matrix-valued trigonometric polynomials with frequencies from g+H. We describe those measures M for which Zg = L2(M) as well as those for which ∩g∈GZg={0} Interpreting M as a spectral measure of a multivariate wide sense stationary process on G and denoting by JH the family of H-cosets, we obtain conditions for JH-singularity and JH-regularity.