1. Berry–Esseen Bounds and Diophantine Approximation
- Author
-
I. Berkes and B. Borda
- Subjects
Uniform distribution (continuous) ,Distribution (number theory) ,Weak convergence ,General Mathematics ,010102 general mathematics ,Diophantine approximation ,Random walk ,Fractional part ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Probability theory ,Irrational number ,0101 mathematics ,Mathematics - 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.
- Published
- 2018
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