Applications of autoreproducing kernel grammian moduli to S (U, H)-valued stationary random functions

AbstractIt is shown that the analytical characterizations of q-variate interpolable and minimal stationary processes obtained by H. Salehi (Ark. Mat., 7 (1967), 305–311; Ark. Mat., 8 (1968), 1–6; J. Math. Anal. Appl., 25 (1969), 653–662), and later by A. Weron (Studia Math., 49 (1974), 165–183), can be easily extended to Hilbert space valued stationary processes when using the two grammian moduli that respectively autoreproduce their correlation kernel and their spectral measure. Furthermore, for these processes, a Wold-Cramér concordance theorem is obtained that generalizes an earlier result established by H. Salehi and J. K. Scheidt (J. Multivar. Anal., 2 (1972), 307–331) and by A. Makagon and A. Weron (J. Multivar. Anal., 6 (1976), 123–137)