Simultaneously Orthogonal Expansion of Two Stationary Gaussian Processes - Examples

Abstract

This paper presents two examples of the simultaneously orthogonal expansion of the sample functions of a pair of stationary Gaussian processes. The pair of Gaussian processes are specified by zero means and covariances exp (–α | s − t |), exp (– β | s − t |) in Example 1 and by 1 − | s − t | /2T, exp (– | s − t | /T) in Example 2. The expansion takes the form of a trigonometric series where the coefficients are mutually independent Gaussian variables for both processes, and the series converges, both with probability one for every t and in the stochastic mean uniformly in t, for both processes. This type of expansion is an extension of the Karhunen-Loéve expansion to the case of a pair of processes, and no concrete example has been given previously. The general theory of the orthogonal expansions is briefly reviewed in Section I, while concrete results for the two examples are tabulated in Section II with a brief outline of the method of derivation. The complete derivation, which constitutes the principal part of this paper, is presented in full detail in Appendices.