On the average product of Gauss-Markov variables

Let xi be members of a stationary sequence of zero mean Gaussian random variables having correlations Exi xj = σ² ρ^|i-j|, 0 < ρ < 1, σ > 0. We address the behavior of the averaged product qm(ρ, σ) ≡ Ex1 x2 ··· x2m−1 x2m as m becomes large. Our principal result when σ² = 1 is that this average approaches zero (infinity) as ρ is less (greater) than the critical value ρc = 0.563007169…. To obtain this we introduce a linear recurrence for the ρm·(ρ, σ), and then continue generating an entire sequence of recurrences, where the (n + 1)-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(ρ, σ). The related problem with qm(ρ, σ) = E| x1 ··· xm| is studied via integral equations and is shown to possess a smaller critical correlation value.