Correlation and spectral theory for periodically correlated random fields indexed on Z2

Abstract

We show that a field X ðm; nÞ is strongly periodically correlated with period ðM; NÞ if and only if there exist commuting unitary operators, U 1 and U 2 that shift the field unitarily by M and N along the respective coordinates. This is equivalent to a field whose shifts on a subgroup are unitary. We also define weakly PC fields in terms of other subgroups of the index set over which the field shifts unitarily. We show that every strongly PC field can be represented as X ðm; nÞ ¼Ũ m 1Ũ n 2 Pðm; nÞ whereŨ 1 andŨ 2 are unitary and Pðm; nÞ is a doubly periodic vectorvalued sequence. This leads to the Gladyshev representations of the field and to strong harmonizability. The 2-and 4-fold Wold decompositions are expressed for weakly commuting strongly PC fields. When the field is strongly commuting, a one-point innovation can be defined. For this case, we give necessary and sufficient conditions for a strongly commuting field to be PC and strongly regular, although possibly of deficient rank, in terms of periodicity and summability of the southwest moving average coefficients. r 2004 Elsevier Inc. All rights reserved.

Information Topological Characterization of Periodically Correlated Processes by Dilation Operators

“…In this paper, the dilation of the spectral measure is presented in the following way. Let {X t } t∈T be a stationary process living on a group T , there is a unitary representation U of T on H X = sp {X t , t ∈ T }, called the time-domain [17], [18], [4], [19] and such that X t = U t X e , where e is identity element of the group T . In this context the operator U is named the shift operator, it describes how to go from any vector X t−1 of H X to X t .…”

Spectrum of periodically correlated fields

“…The above description of the SO-spectrum of a PC process, which originates from Gladyshev's papers [12,13], can be easily extended to the case of coordinate-wise strongly periodically correlated fields over R n or Z n (see e.g. [1,7,6,11,21]). The purpose of this work is to describe the SOspectrum of a K-periodically correlated field for any closed subgroup K of an LCA group G and as a particular case when G = R m × Z n .…”

“…Example 2 Let T and S be two non-zero integers. Suppose that the field X on Z 2 is weakly PC with period (T, S) ( [21]), that is K X (m, n), (u, v) = K X (m + T, n + S), (u + T, v + S) , for all n, m, u, v ∈ Z Here K = k(T, S) : k ∈ Z . We assume that at least one of T or S is positive.…”

“…A summary of the theory of PC sequences can be found in [23]. Surprisingly, there are only several works [2,3,4,6,7,10,11,21,36] dealing with PC fields, and each one concentrates on a particular type, namely coordinate-wise strong periodicity. An intention of this paper is to sketch a unified theory of fields over any locally compact Abelian (LCA) group G which are periodically correlated with respect to an arbitrary closed subgroup K of G. We emphasize the case of G = Z n ×R m to illustrate the results.…”

“…The last section contains examples that illustrate the theory developed. In particular Example 2 gives a complete analysis of the weakly periodically correlated fields over Z 2 , introduced in [21].…”