Increasing permutations of random processes

Abstract

The paper is concerned with an application of limit theorems to the study of increasing permutations of stable random processes. By the increasing permutation of a function is meant the nondecreasing function with the same distribution. The trajectories of a random process may be approximated by step-functions, and then the continuity of the increasing permutation operator permits one to apply the Skorokhod invariance principle to obtain the distribution of the random process. The distribution function and the expected value of the increasing permutation of a stable random process are given explicitly. Also the univariate distributions of the increasing permutation of the Cauchy process are obtained. In various normed spaces the images of the unit balls with respect to the operator of increasing permutation are described. A separate section is devoted to the increasing permutations of higher-dimensional processes. Bibliography: 5 titles.