Completely regular multivariate stationary processes and the Muckenhoupt condition

Abstract

In this paper we shall give a necessary and sufficient condition for a multivariate stationary stochastic process to be completely regular. Let us recall that a multivariate stationary stochastic process with discrete time 1 is a sequence of d-tuples x(n) = (x1(n), x2(n), . . . , xd(n)), n ∈ Z of scalar random variables such that E|xj(n)| <∞ and the correlation matrix Q(n, k) Q(n, k) = {Q(n, k)i,j}1≤i,j≤d := { Exi(n)xj(k) }