JH-singularity and JH-regularity of multivariate stationary processes over LCA groups

LUTZ PETER KLOTZ; JUAN MIGUEL MEDINA

PROBABILITY AND MATHEMATICAL STATISTICS

Urbanik Center for Probability and Mathematical Statistics//University of Wrocław

Lugar: Wroclaw; Año: 2021 vol. 41 p. 173 - 192

0208-4147

Let G be an LCA group, 􀀀 its dual group, H a closed subgroup of Gsuch that its annihilator is countable. Let M denote a regular positivesemidefinite matrix-valued Borel measure on 􀀀 and L2(M) the correspondingHilbert space of matrix-valued functions square-integrable with respect toM.For g 2 G, let Zg be the closure in L2(M) of all matrix-valued trigonometricpolynomials with frequencies from g+H. We describe those measures M forwhich Zg = L2(M) as well as those for which Tg2G Zg = {0}. InterpretingM as a spectral measure of a multivariate wide sense stationary processon G and denoting by JH the family of H-cosets this way conditions forJH-singularity and JH-regularity, respectively, of the process are obtained.