On the Average Product of Gauss‐Markov Variables

Let x_i be members of a stationary sequence of zero mean gaussian random variables having correlations Ex_ix_j = σ²ρ|i−j|, 0< ρ <, σ > 0. We address the behavior of the averaged product q_m(ρ, σ) ≡ Ex₁x₂ … x_2m−1 x_2m 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 ρ_c = 0.563007169 … To obtain this we introduce a linear recurrence for the q_m. (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_m(ρ, σ). The related problem with q_m (ρ, σ) = E|x₁ … x_m| is studied via integral equations and is shown to possess a smaller critical correlation value.