Further decomposition of the Karhunen-Loève series representation of a stationary random process

It is shown how the Karhunen-Loève series representation for a finite sample of a discrete random sequence, stationary to the second order, may be further decomposed into a pair of series by utilizing certain symmetry properties of the covariance matrix of the sequence. The theory is applied to the particular example of a first-order Markov sequence, the series representation of which has not so far been reported in the literature. The generalization to the case of continuous random functions on a finite interval is similar and is therefore only briefly described.