TY - JOUR

T1 - On the Average Product of Gauss‐Markov Variables

AU - Logan, B. F.

AU - Mazo, J. E.

AU - Odlyzko, A. M.

AU - Shepp, L. A.

PY - 1983/12

Y1 - 1983/12

N2 - 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.

AB - 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.

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U2 - 10.1002/j.1538-7305.1983.tb03463.x

DO - 10.1002/j.1538-7305.1983.tb03463.x

M3 - Article

AN - SCOPUS:0020888949

SN - 0005-8580

VL - 62

SP - 2993

EP - 3006

JO - Bell System Technical Journal

JF - Bell System Technical Journal

IS - 10

ER -