On the Average Product of Gauss-Markov Variables
B. F. Logan, James E. Mazo, Andrew M. Odlyzko, Larry A Shepp · Bell System Technical Journal · 1983
Let xibe members of a stationary sequence of zero mean Gaussian random variables having correlations Exixj= σ2ρ|i-j|, 00. We address the behavior of the averaged product qm(ρ, σ) ≡ Ex1x2··· x2m−1x2mas 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.