On the Average Product of Gauss‐Markov Variables

B. F. Logan, J. E. Mazo, A. M. Odlyzko, L. A. Shepp

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Abstract

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 languageEnglish (US)
Pages (from-to)2993-3006
Number of pages14
JournalBell System Technical Journal
Volume62
Issue number10
DOIs
StatePublished - Dec 1983

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