Mutual subordination of multivariate stationary processes over any locally compact abelian group

Summary Our purpose is to extend Kolmogorov's theorem 5, Th. 10] on mutual subordination for univariate weakly stationary stochastic processes over the (discrete) group of integers to multivariate processes over any (Hausdorff) locally compact abelian (lca) group. This extension is given in Theorems (1.12) and (3.4) below. We shall lean heavily on the joint paper 10] on the decomposition of matricial measures, to which the present paper may be regarded as a sequel.In Section 1 of the paper we shall define and prove theorems on the concept of E-subordination, where E is a projection-valued measure. In Section 2 we shall examine the structure of stationary processes over an lca group. In Section 3 we shall consider the concept of subordination of stationary processes. Finally in Section 4, we shall apply our subordination theorems to deduce that matrixvalued functions in L ₂ on the unit circle having no negative frequencies have a constant rank a.e. (Lebesgue) (4.2), a theorem of F. and M. Riesz (4.3), and a theorem on wandering subspaces due to Robertson 9] (4.4).The author is grateful to Prof. P. R. Masani for his generous help in formulating this paper.