Stationary Processes as Shifts of Functions of Independent Random Variables

Abstract

1, Summary. 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 £„ = g(xn , xn~t , • • •)> & function of xn , xn„x, • • • , that is inde­ pendent of the past, that is, independent of xn-x , £w_2 , • • • . Such a sequence of random variables {£n} is a sequence of independent and identically distributed random variables. Further, given such a construction, is xn a function of ?» , £n_i , • • • . Necessary and sufficient conditions for such a representation are ob­ tained 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 = (ptJ) 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 £n's be functions of xn , xn-x , • • • is removed. We ask whether for some such family {£n} there is a process {yn}, yn = gr(£n f g n- 1 f ...) , 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.