Measurable approximation of a second-order process

Abstract

An important problem, having practical as well as theoretical significance, is the problem of determining whether or not a stochastic process has a measurable modification or can be approximated (in the mean square sense) by a second-order process admitting such a modification. It is known [1] that a second-order process has a measurable modification if and only if its covariance function is measurable and its linear space is separable. However, a problem still open is whether sufficient conditions for the existence of a measurable modification can be given in terms of the covariance function only, so that they have a pure analytical form. In this paper we formulate one set of such conditions (Theorem 2, Corollary 2.2) and from that derive a result giving conditions under which a second-order process can be uniformly approximated (in the mean square sense) by a measurable second-order process (Theorem 3).