Limit distributions in metric discrepancy theory
type: Item
instance of: scholarly article
title: Limit distributions in metric discrepancy theory
zbMATH Open document ID: 1287.11094
author: István Berkes
author: Christoph Aistleitner
published in: Monatshefte für Mathematik
publication date: 2013-03-18T00:00:00Z
review text: The authors consider lacunary sequences \((n_k)_{k\geq 1}\) of integers, i.e. sequences satisfying Hadamard's gap condition \(n_{k+1}/n_k\geq q > 1,\quad (k=1,2,\ldots)\). In the main result the limit distribution of \(\sqrt{N}D_N (n_kx)\) (for \(N\to\infty)\) is determined, where \(D_N(n_kx)\) denotes the discrepancy of the sequence \((n_kx)_{k\geq 1}\bmod 1\). The result holds provided that the number of solutions \((k,l)\) of the Diophantine equation \(an_k- bn_l= c\) with \(1\leq k\), \(l \leq N\) is ``small''. The limit distribution is a so-called Kolmogorov distribution with explicitly known densities. The special case \(n_k= a^k\) is studied in more detail and extensions to \(L^2\)-discrepancies are also derived.
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 6146035
zbMATH Keywords: lacunary series
zbMATH Keywords: discrepancy
zbMATH Keywords: covariance kernel
zbMATH Keywords: Brownian bridge
reviewed by: Robert F. Tichy
MaRDI profile type: Publication
full work available at URL: s00605-012-0378-9
OpenAlex ID: W2070520807
cites work: Random Fields and Geometry
cites work: Irregular discrepancy behavior of lacunary series
cites work: Irregular discrepancy behavior of lacunary series. II
cites work: On the law of the iterated logarithm for the discrepancy of lacunary sequences
cites work: On the central limit theorem for \(f(n_{k}x)\)
cites work: Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes
cites work: The Size of Trigonometric and Walsh Series and Uniform Distribution Mod 1
cites work: Q5560061
cites work: The Asymptotic Distribution of the Range of Sums of Independent Random Variables
cites work: The law of the iterated logarithm for discrepancies of \(\{\theta^{n}x\}\)
cites work: On the distribution of values of sums of the type \(\sum f(2^kt)\)
cites work: Limit theorems for lacunary series and uniform distribution mod 1
cites work: A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables
cites work: Q3395931
cites work: Q4039793
DOI: 10.1007/S00605-012-0378-9
description: scientific article; zbMATH DE number 6146035