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Vestnik St. Petersburg University, Mathematics

, Volume 51, Issue 4, pp 343–359 | Cite as

On the History of the St. Petersburg School of Probability and Statistics. III. Distributions of Functionals of Processes, Stochastic Geometry, and Extrema

  • A. N. BorodinEmail author
  • Yu. A. Davydov
  • V. B. Nevzorov
Mathematics
  • 1 Downloads

Abstract

This is the third paper in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg School of Probability and Statistics in 1947–2017. The paper deals with the studies on functionals of random processes, some problems of stochastic geometry, and problems associated with ordered systems of random variables. The first sections of the paper are devoted to the problems of calculating the distributions of various functionals of Brownian motion and consider the so-called invariance principles for Brownian local times and random walks. The second part is dedicated to limit theorems for weakly dependent random variables and local limit theorems for stochastic functionals. It provides information about the stratification method and the local invariance principle. The asymptotic behavior of the convex hulls of random samples of increasing size and limit theorems for random zonotopes are also considered. An important relation between Poisson point processes and stable distributions is explained. The final part presents extensive information on research related to ordered systems of random variables. The maxima of sequential sums, order statistics, and record values are analyzed in detail.

Keywords

Brownian motion distribution of functionals Brownian local time random walks invariance principle stratification method local invariance principle limit theorems for random zonotopes and convex hulls order statistics extrema records 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • A. N. Borodin
    • 1
    • 2
    Email author
  • Yu. A. Davydov
    • 1
  • V. B. Nevzorov
    • 1
  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Steklov Mathematical Institute, Russian Academy of SciencesSt. Petersburg BranchSt. PetersburgRussia

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