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

We show that a field X(m,n) is strongly periodically correlated with period (M,N) if and only if there exist commuting unitary operators, U1 and U2 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(m,n)=Ũ1mŨ2nP(m,n) where Ũ1 and Ũ2 are unitary and P(m,n) is a doubly periodic vector-valued 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.