Increasing permutations of random processes

E. E. Zhukova

Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part 13, Tome 216 (1994), pp. 62-75

Résumé

The paper concerns with the application of limit theorems to the study of increasing permutations of stable random processes. By an increasing permutation of a function a non-decreasing function with the same distribution is meant. The trajectory of a random process may be approximated by a step-function, and then the continuity of the increasing permutation operator permits to apply the Skorohod 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 one-dimensional distribution of the increasing permutation of the Cauchy process is obtained. In various normed spaces the images of unit balls with respect to the operator of increasing permutation are determined. A separate section is devoted to the increasing permutations of higher dimensional processes. Bibliography: 5 titles.