The Annals of Statistics

Unit Canonical Correlations between Future and Past

E. J. Hannan and D. S. Poskitt

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Stationary vector ARMA processes $x(t), t = 0, \pm 1, \pm 2, \cdots$, of $n$ components are considered that are of full rank, and the situation where there are linear functions of the future $x(t), t > 0$, and the past $x(t), t \leq 0$ (more properly the present and the past) that have unit correlation. It is shown that the number of linearly independent such pairs (i.e., the number of unit canonical correlations between future and past) is the number of zeros of the determinant of the transfer functions, from innovations to outputs, that lie on the unit circle, counting these with their multiplicities.

Article information

Ann. Statist., Volume 16, Number 2 (1988), 784-790.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 60G10: Stationary processes
Secondary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]

Stationary vector process ARMA process canonical correlation Kronecker indices Hardy spaces


Hannan, E. J.; Poskitt, D. S. Unit Canonical Correlations between Future and Past. Ann. Statist. 16 (1988), no. 2, 784--790. doi:10.1214/aos/1176350836.

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