Stochastic Variable Theory

Abstract

Let X(t) be a random function of time t, where t Є [a,b], − ∞ < a < b < ∞ or t Є [a; [),or t Є (−∞, ∞), as the specific case dictates. One way to describe the observable properties of the set of random functions, a.k.a. (also known as) a stochastic variable, is by means of a collection of correlation functions

$${C}_{l}({t}_{1}, {t}_{2}, \ldots , {t}_{l})\equiv \langle X ({t}_{1}) X ({t}_{2}) \ldots X ({t}_{l}) \rangle, $$

for all $l \geq 1$ If the functions in the set {C _l } are pointwise defined, then X(l) is called a stochastic process. On the other hand, if the functions in the set are distributional in nature, then X(l is called a generalized stochastic process. We will have occasion to discuss both types.