Geometric problems in the theory of infinite-dimensional probability distributions

the long version

This monograph is a collection of some results, published previously but mostly without detailed proofs, of investigations in the theory of probability distributions. Of the three chapters making up this book, the first contains only auxiliary information. The division of the whole content into two chapters (II and Ill) corresponds to the two main themes of the investigations. The second and third chapters have in common the geometric character of the problems studied and the related definite unity of methods, although in content these chapters are formally independent of each other. A study of the measure of a solid in a Hilbert space occupies one of the central places in the second chapter, which deals with the properties of sample functions of random processes; the problem of the properties of the sample functions is studied in terms of the geometry of a certain subset of the Hilbert space determined by the random process and the relevant property of the realization. On the other hand, the basic theorem of the third chapter, whose proof makes up the content of §10, can be naturally stated as a result on the extreme points of the infinite-dimensional analogue of the so-called "Hungarian polyhedron" determined by specifying the marginal distributions of two statistics (two measurable decompositions) and con- Sisting Of all the measures having the given marginal distributions. The remaining sections of the third chapter are closely related in substance to this result or are directly based on it.