Berry–Esseen Bounds and Diophantine Approximation

I. Berkes; B. Borda

doi:10.1007/s10476-018-0203-3

Berry–Esseen Bounds and Diophantine Approximation

I. Berkes^*, B. Borda

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

Let S_N, N = 1, 2,.. be a random walk on the integers, let α be an irrational number and let Z_N = {S_Nα}, where {·} denotes fractional part. Then Z_N, N = 1, 2,.. is a random walk on the circle, and from classical results of probability theory it follows that the distribution of Z_N 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 S_N, depends sensitively on the rational approximation properties of α.

Original language	English
Pages (from-to)	149-161
Number of pages	13
Journal	Analysis Mathematica
Volume	44
Issue number	2
DOIs	https://doi.org/10.1007/s10476-018-0203-3
Publication status	Published - 1 Jun 2018
Externally published	Yes

Keywords

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

ASJC Scopus subject areas

Mathematics(all)

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10.1007/s10476-018-0203-3

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AU - Borda, B.

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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

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Berry–Esseen Bounds and Diophantine Approximation

Abstract

Keywords

ASJC Scopus subject areas

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