Spectral distance measures between Gaussian processes

Abstract

Utilizing asymptotic results from prediction theory of multivariate stationary random processes and from regression theory for multivariate stationary processes, we develop asymptotic (large sample) expressions for the Chernoff coefficient, Bhattacharyya distance, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</tex> -divergence and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</tex> -divergence between two <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> -dimensional, covariance stationary Gaussian processes on the basis of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> discrete-time samples. The expressions are given in terms of the two spectral density matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F_{1}(\lambda), F_{2}(\lambda)</tex> derived from the two autocovariance matrix sequences, and of the spectral density matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M(\lambda)</tex> related to the sequence of differences of mean vectors. The resulting spectral expressions are useful in a variety of applications, as discussed in the paper.