Open AccessOn bounds for convergence rates in combinatorial strong limit theorems and its applications
Author(s) -
Alexander O. Frolov
Publication year - 2020
Publication title -
vestnik sankt-peterburgskogo universiteta. matematika. mehanika. astronomiâ/vestnik sankt-peterburgskogo universiteta. seriâ 1, matematika, mehanika, astronomiâ
Language(s) - English
Resource type - Journals
eISSN - 2587-5884
pISSN - 1025-3106
DOI - 10.21638/spbu01.2020.410
Subject(s) - mathematics , law of the iterated logarithm , combinatorics , permutation (music) , random permutation , logarithm , rank (graph theory) , law of large numbers , rate of convergence , limit (mathematics) , central limit theorem , matrix (chemical analysis) , distribution (mathematics) , iterated function , convergence of random variables , order (exchange) , random variable , iterated logarithm , convergence (economics) , statistics , symmetric group , mathematical analysis , channel (broadcasting) , physics , materials science , economic growth , acoustics , economics , composite material , finance , electrical engineering , engineering
We find necessary and sufficient conditions for convergences of series of weighted probabilities of large deviations for combinatorial sums i Xniπn(i), where Xnij is a matrix of order n of independent random variables and (πn(1), πn(2), . . . , πn(n)) is a random permutation with the uniform distribution on the set of permutations of numbers 1, 2, . . . , n, independent with Xnij. We obtain combinatorial variants of results on convergence rates in the strong law of large numbers and the law of the iterated logarithm under conditions closed to optimal ones. We discuss applications to rank statistics.