Sums of stationary random variables

Abstract

A sequence x(t) (-oo <t < o, t an integer) of elements in Hilbert space is called stationary if the inner product (x(t+s), x(t)) does not depend upon t. If the Hilbert space is L2 space with probability measure, then x(t) is a random variable and the sequence x(t) (X <t < oo ) is called a second-order stationary random process. Let X be the closed linear manifold spanned by all the elements of the stationary process. Then Kolmogorov [I] has shown that the equation x(t) U =x (t +1), o <t < oo, uniquely determines the unitary operator U with domain and range X. Using the von Neumann [2] spectral representation of U, we obtain the spectral representation of the random process