The measurability of a stochastic process of second order and its linear space

Abstract

It is of considerable theoretical and practical interest to know whether a stochastic process has a measurable modification. For the important class of second order processes, simple necessary and sufficient conditions for the existence of a measurable modification are given in terms of the autocorrelation of the process and the separability of its reproducing kernel Hilbert space or its linear space. It is shown that weakly continuous processes, processes with orthogonal increments and second order martingales always have measurable modifications. Also necessary and sufficient conditions are given in terms of integral representations for the linear space of a second order process to be separable. As a consequence it is shown that a second order process is oscillatory if and only if its linear space is separable.