Berry–Esseen Bounds and Diophantine Approximation

I. Berkes*, B. Borda

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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
Issue number2
Publication statusPublished - 1 Jun 2018
Externally publishedYes


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

ASJC Scopus subject areas

  • Mathematics(all)

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