## The Annals of Statistics

### Unit Canonical Correlations between Future and Past

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176350836

Digital Object Identifier
doi:10.1214/aos/1176350836

Mathematical Reviews number (MathSciNet)
MR947578

Zentralblatt MATH identifier
0646.62082

JSTOR