On the Average Product of Gauss-Markov Variables

On the Average Product of Gauss-Markov Variables By B. F. LOGAN, Jr.,* J. E. MAZO,* A. M. ODLYZKO* and L. A. SHEPP* (Manuscript received April 8, 1983) Let Xi be members of a stationary sequence of zero mean gaussian random variables having correlations Ex,xj = a2pu~A, 0 p 1, a > 0. We address the behavior of the averaged product qm(p, a) = ExiX2 · · · x2m-x2m as m becomes large. Our principal result when a2 = 1 is that this average approaches zero (infinity) as p is less (greater) than the critical value pc = 0.563007169 To obtain this we introduce a linear recurrence for the qm-(p, a), and then continue generating an entire sequence of recurrences, where the (n -I- l)-st relation is a recurrence for the coefficients that appear in the nth relation. This leads to a new, simple continued fraction representation for the generating function of the qm(p, ff). The related problem with qm(p, a) = E Xi · · · xm | is studied via integral equations and is shown to possess a smaller critical correlation value. I. INTRODUCTION The problem that we consider in this paper is as follows: Let {x.-JT be a stationary sequence of zero mean, gaussian random variables with covariances Pu = E x ^ = aV'WI, 0 p 1, a > 0; i , j = 1, 2, . . . , (1)