On the History of St. Petersburg School of Probability and Mathematical Statistics: II. Random Processes and Dependent Variables

• Published in: Vestnik, St. Petersburg University. Mathematics Vol. 51; no. 3; pp. 213 - 236
• Main Authors: Ibragimov, I. A., Lifshits, M. A., Nazarov, A. I., Zaporozhets, D. N.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.07.2018; Springer Nature B.V
• Subjects: Analysis; Approximation; Dependent variables; Gaussian process; Markov chains; Martingales; Mathematics; Mathematics and Statistics; Normal distribution; Norms; Probabilistic models; Probability theory; Random processes; Random variables; Sequences; Statistical analysis; sticky particle systems; approximation of processes of growing dimension; limit theorem for sums of dependent variables; Gaussian processes; functional law of iterated logarithm; small deviation probabilities
• ISSN: 1063-4541; 1934-7855
• DOI: 10.3103/S1063454118030123

This is the second paper in a series of reviews devoted to the scientific achievements of the Leningrad and St. Petersburg school of probability and mathematical statistics from 1947 to 2017. This paper is devoted to the works on limit theorems for dependent variables (in particular, Markov chains, sequences with mixing properties, and sequences admitting a martingale approximation) and to various aspects of the theory of random processes. We pay particular attention to Gaussian processes, including isoperimetric inequalities, estimates of the probabilities of small deviations in various norms, and the functional law of the iterated logarithm. We present a brief review and bibliography of the works on approximation of random fields with a parameter of growing dimension and probabilistic models of systems of sticky inelastic particles (including laws of large numbers and estimates for the probabilities of large deviations).