On modifications of random processes

Abstract

UDC 519.2 0. In the paper, some questions related to the existence, canonicity, and properties of versions and modifications of a random process are discussed. Notions of oscillations (in several meanings) of a random process are introduced. Some properties of modifications of a random process ave characterized in terms of oscillations. Methods for constructing the natural modification (in the sense of B. S. Tsirel'son [1]) are considered. The notion of the normal modification of a process is defined, and its properties are established. A simple criterion of measurability is proved for linear functionals on a Gaussian measure vector space. 1. The term "random process" can be understood in different senses. Therefore, in discussing modifications of random processes, one should first of all make the terminology more precise. The word "process" induces the idea of something developing in time. This is not always desirable, because often merely a set K of random variables is under consideration. Still the term "set" is too broad, and the term "family" makes one concentrate too much on parametric set. The main idea of the approach elaborated in this paper (as well as in many papers of many other authors dealing with the theory of Gaussian families such as, say, the paper [1] cited above) consists in totally ignoring all additional structures. Here "additional" means "noncanonical," i.e., such that cannot be restored uniquely, given finite-dimensional distributions. Maybe one had better speak about "random fields," and some authors do so. Still the word "field" also generates undesirable associations as if the parametric set were finite-dimensional or a group. Taking all this into account, we retain the term "random process." Often we shall consider a random process K as a family with self-parametrization: each of its elements will be simultaneously considered as a (corresponding to itself) point of the parametric set K.