To the history of Saint-Petersburg school of Probability and Statistics. III. Functional distributions, stochastic geometry and extrema

Abstract

This is the third article in a series of surveys dedicated to the scientific achievements of the Leningrad —Saint-Petersburg school of Probability and Statistics in the period from 1947 to 2017. It is devoted to some studies of functionals of random processes, some problems of stochastic geometry and some questions connected with ordered systems of random variables. In the first section of the article, we talk about problems of calculation of distributions for various functionals of Brownian motion. The so-called invariance principles are also considered for Brownian local times and random walks. The second section is devoted to limit theorems for weakly dependent random variables and to local limit theorems for stochastic functionals. A method of stratifications and the local invariance principle are also mentioned. In the following section, we consider the asymptotic behavior of convex hulls of random samples and limit theorems for random zonotopes. An important relation between Poisson point processes and stable distributions is discussed. In the final part, we provide exhaustive information on research related to ordered systems of random variables. The maxima of consecutive sums, order statistics and record values are considered in detail.