### 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_{N}α>}, 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 | |

Publication status | Published - Jun 1 2018 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Analysis Mathematica*,

*44*(2), 149-161. https://doi.org/10.1007/s10476-018-0203-3

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

Research output: Contribution to journal › Article

*Analysis Mathematica*, vol. 44, no. 2, pp. 149-161. https://doi.org/10.1007/s10476-018-0203-3

}

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 -