On Separability of Families of Reversed Submartingales

Abstract

Certain statistical models are described by a family of reversed submartingales. The main problem under consideration is to estimate maximum points of the associated information function. It turns out that some maximal inequalities are important to be established in this direction. In this paper we show that these inequalities remain valid for suitable chosen modifications of those families. However this result has no practical meaning for statistics. For that reason we introduce and investigate a concept of separability of the family of reversed submartingales which is according to [3] indexed by an analytic metric space. We prove the existence of separable modifications for these families and show that semicontinuity implies separability. Our method in these considerations relies upon results of the classical theory of stochastic processes. Finally we introduce conditionally S-regular families of reversed submartingales as those which are as close as possible to satisfy the maximal inequalities. We show that separable families belong to this class. Moreover we present one significant class of this type which includes all U-processes and cover a large number of random functions occurring in probability and statistics.