Let xi be members of a stationary sequence of zero mean gaussian random variables having correlations Exixj = σ2ρ|i−j|, 0< ρ <, σ > 0. We address the behavior of the averaged product qm(ρ, σ) ≡ Ex1x2 … 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 qm. (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.
|Original language||English (US)|
|Number of pages||14|
|Journal||Bell System Technical Journal|
|State||Published - Dec 1983|