## The Annals of Mathematical Statistics

### On the Multivariate Analysis of Weakly Stationary Stochastic Processes

L. H. Koopmans

#### Abstract

The existence of the class of orthogonal projections which map an arbitrary $q$-variate weakly stationary stochastic process again into a $q$-variate process contained in the span of $p(\leqq q)$ of its component processes is established. Mimicking the definitions of the partial and multiple correlation coefficients (e.g., Anderson, 1958), these projections are used to define partial and multiple coefficients of coherence, thus providing the foundation for the multivariate covariance and correlation analyses for weakly stationary processes employed in special cases by Tick (1963) and Jenkins (1963). Some of the properties of the partial and multiple correlation coefficients are established for the corresponding coefficients of coherence. In particular, formulas are established for generating these parameters iteratively. When used for the sample coefficients of coherence, these formulas provide useful methods of defining and constructing estimates of the multiple and partial coefficients of coherence from the usual estimates of the ordinary coefficient of coherence. Results due to Goodman (1963) concerning the distributions of these estimators when the process is Gaussian are indicated.

#### Article information

Source
Ann. Math. Statist., Volume 35, Number 4 (1964), 1765-1780.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177700398

Digital Object Identifier
doi:10.1214/aoms/1177700398

Mathematical Reviews number (MathSciNet)
MR170380

Zentralblatt MATH identifier
0132.13304

JSTOR