Stationary Processes as Shifts of Functions of Independent Random Variables

Abstract

Let xn, n = 0, ±1, ±2, …, be a strictly stationary process. Two closely related problems are posed with respect to the structure of strictly stationary processes. In the first problem we ask whether one can construct a random variable ξn = g(xn, xn-1, …), a function of xn, xn-1, …, that is independent of the past, that is, independent of xn-1, xn-2, …. Such a sequence of random variables {ξn} is a sequence of independent and identically distributed random variables. Further, given such a construction, is xn a function of ξn, ξn-1, …. Necessary and sufficient conditions for such a representation are obtained in the case where xn is a finite state Markov chain with the positive transition probabilities in any row of the transition probability matrix P = (pii) of xn distinct (Section 3). Such a representation is comparatively rare for a finite state Markov chain. In the second problem, the assumption that the independent and identically distributed ξn’s be functions of xn, xn-1, … is removed. We ask whether for some such family {ξn} there is a process {yn}, yn = g(ξn, ξn-1,…), with the same probability structure as {xn}. This is shown to be the case for every ergodic finite state Markov chain with nonperiodic states (Section 4). Sufficient conditions for such representations in the case of a general strictly stationary process are obtained in Section 5.