Berry–Esseen Bounds and Diophantine Approximation

I. Berkes, B. Borda

Research output: Contribution to journalArticle

Abstract

Let SN, N = 1, 2,.. be a random walk on the integers, let α be an irrational number and let ZN = {SNα>}, where {·} denotes fractional part. Then ZN, N = 1, 2,.. is a random walk on the circle, and from classical results of probability theory it follows that the distribution of ZN converges weakly to the uniform distribution. We determine the precise speed of convergence, which, in addition to the distribution of the elementary step X of the random walk SN, depends sensitively on the rational approximation properties of α.

Original languageEnglish
Pages (from-to)149-161
Number of pages13
JournalAnalysis Mathematica
Volume44
Issue number2
DOIs
Publication statusPublished - Jun 1 2018

Fingerprint

Berry-Esseen Bound
Diophantine Approximation
Random walk
Fractional Parts
Irrational number
Rational Approximation
Speed of Convergence
Approximation Property
Probability Theory
Uniform distribution
Circle
Denote
Converge
Integer

Keywords

  • convergence speed
  • Diophantine approximation
  • i.i.d. sums mod 1
  • weak convergence

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Berry–Esseen Bounds and Diophantine Approximation. / Berkes, I.; Borda, B.

In: Analysis Mathematica, Vol. 44, No. 2, 01.06.2018, p. 149-161.

Research output: Contribution to journalArticle

Berkes, I. ; Borda, B. / Berry–Esseen Bounds and Diophantine Approximation. In: Analysis Mathematica. 2018 ; Vol. 44, No. 2. pp. 149-161.
@article{d57ac85cf8634945bfe2c1c400251298,
title = "Berry–Esseen Bounds and Diophantine Approximation",
abstract = "Let SN, N = 1, 2,.. be a random walk on the integers, let α be an irrational number and let ZN = {SNα>}, where {·} denotes fractional part. Then ZN, N = 1, 2,.. is a random walk on the circle, and from classical results of probability theory it follows that the distribution of ZN converges weakly to the uniform distribution. We determine the precise speed of convergence, which, in addition to the distribution of the elementary step X of the random walk SN, depends sensitively on the rational approximation properties of α.",
keywords = "convergence speed, Diophantine approximation, i.i.d. sums mod 1, weak convergence",
author = "I. Berkes and B. Borda",
year = "2018",
month = "6",
day = "1",
doi = "10.1007/s10476-018-0203-3",
language = "English",
volume = "44",
pages = "149--161",
journal = "Analysis Mathematica",
issn = "0133-3852",
publisher = "Springer Netherlands",
number = "2",

}

TY - JOUR

T1 - Berry–Esseen Bounds and Diophantine Approximation

AU - Berkes, I.

AU - Borda, B.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - Let SN, N = 1, 2,.. be a random walk on the integers, let α be an irrational number and let ZN = {SNα>}, where {·} denotes fractional part. Then ZN, N = 1, 2,.. is a random walk on the circle, and from classical results of probability theory it follows that the distribution of ZN converges weakly to the uniform distribution. We determine the precise speed of convergence, which, in addition to the distribution of the elementary step X of the random walk SN, depends sensitively on the rational approximation properties of α.

AB - Let SN, N = 1, 2,.. be a random walk on the integers, let α be an irrational number and let ZN = {SNα>}, where {·} denotes fractional part. Then ZN, N = 1, 2,.. is a random walk on the circle, and from classical results of probability theory it follows that the distribution of ZN converges weakly to the uniform distribution. We determine the precise speed of convergence, which, in addition to the distribution of the elementary step X of the random walk SN, depends sensitively on the rational approximation properties of α.

KW - convergence speed

KW - Diophantine approximation

KW - i.i.d. sums mod 1

KW - weak convergence

UR - http://www.scopus.com/inward/record.url?scp=85048526316&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85048526316&partnerID=8YFLogxK

U2 - 10.1007/s10476-018-0203-3

DO - 10.1007/s10476-018-0203-3

M3 - Article

VL - 44

SP - 149

EP - 161

JO - Analysis Mathematica

JF - Analysis Mathematica

SN - 0133-3852

IS - 2

ER -