ABSTRACT
One of the most elegant uses of orthogonal systems is in conjunction with continuous stochastic processes. Such a process {X(t),t G / } is a family of random variables indexed by the parameter t [Ro, p.91]. For processes that are continuous in mean square, i.e.,
E\X(t)\2 < oo
and
E\X(t) -X(s)\2 ->0 as
there is a theory, the Karhunen-Loeve theory, that converts the process into an uncorrelated sequence of random variables. This is done by using a certain orthogonal system, expanding X(t) in terms of i t , and showing the coefficients are uncorrelated.
