Some Novel Properties of Wiener's Canonical Expansion

Abstract

An expression of Wiener's orthogonal functionals {Fn} for an arbitrary variance parameter is presented along with some of its salient properties. A necessary and sufficient condition for the convergence of the corresponding Wiener series is given. Introduction of the variance parameter enables us to discuss some interesting algebraic properties of Wiener's canonical networks also. It is shown that they form a Boolean algebra, each element of which represents a whole family of Wiener's canonical networks for a fixed variance parameter. Applicability of the Wiener theory to time-variable systems and to non-Gaussian processes is briefly discussed.