On the average product of Gauss-Markov variables

Let x<inf>i</inf> be members of a stationary sequence of zero mean Gaussian random variables having correlations Ex<inf>i</inf> x<inf>j</inf> = σ<sup>2</sup> ρ<sup>|i-j|</sup>, 0 < ρ < 1, σ > 0. We address the behavior of the averaged product q<inf>m</inf>(ρ, σ) ≡ Ex<inf>1</inf> x<inf>2</inf> ··· x<inf>2m−1</inf> x<inf>2m</inf> as m becomes large. Our principal result when σ<sup>2</sup> = 1 is that this average approaches zero (infinity) as ρ is less (greater) than the critical value ρ<inf>c</inf> = 0.563007169…. To obtain this we introduce a linear recurrence for the ρ<inf>m</inf>·(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 q<inf>m</inf>(ρ, σ). The related problem with q<inf>m</inf>(ρ, σ) = E| x<inf>1</inf> ··· x<inf>m</inf>| is studied via integral equations and is shown to possess a smaller critical correlation value.