Some Remarks on an Interpolation Problem of A. M. Yaglom

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

“…However, the formulation of the result as well as the proof given there were somewhat imprecise or even misleading and Theorem 2.2 removes this inaccuracy. Moreover, [6] was written ignoring Pourahmadi's paper [14]. Here we point out the interrelations between Pourahmadi's results and ours.…”

“…Section 5 deals with the description of the spectral measure of a J H -regular process. Among other things, we again generalize and improve a result of [6]. Already the early papers on stationary time series dealt with the Wold decomposition of a stationary process into its J -regular and J -singular parts: see [20], [10], [22] for processes on Z or R and the family J := {(−∞, x] ∩ Z : x 0} or J := {(−∞, x] : x 0}; Salehi and Scheidt [18,Theorem 2.13] proved the existence and uniqueness of a Wold decomposition for processes on an LCA group G and a family J of subsets of G invariant under translation.…”

“…This method has the advantage that the resulting measureμ is finite, whereas ν is only σ-finite in general. Yaglom's approach was used in [7] and generalized to multivariate processes in [17] and [6] as well as to harmonizable symmetric α-stable processes in [9].…”

“…On the basis of abstract harmonic analysis we gain a unified approach to stationary processes on various groups. Note that in [6] processes on R and Z were discussed separately. Since our results can be applied, for example, to a homogeneous field on a torus or cylinder, they are also of practical interest.…”

“…Then Theorem 2.2 gives a complete characterization of the spectral measures of J H -singular processes. The special case G = R and H = Z was solved in [6]. However, the formulation of the result as well as the proof given there were somewhat imprecise or even misleading and Theorem 2.2 removes this inaccuracy.…”