On the discrepancy of random subsequences of [n alpha}, II

Berkes, István and Borda, Bence (2021) On the discrepancy of random subsequences of [n alpha}, II. ACTA ARITHMETICA, 199 (3). pp. 303-330. ISSN 0065-1036

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Abstract

Let α be an irrational number, let X1,X2,... be independent, identically distributed, integer-valued random variables, and put Sk = k j=1Xj. Assuming that X1 has finite variance or heavy tails P(|X1| > t) ∼ ct−β, 0 < β < 2, in [4] we proved that, up to logarithmic factors, the order of magnitude of the discrepancy DN(Skα) of the first N terms of the sequence {Skα} is O(N−τ), where τ = min(1/(βγ),1/2) (with β = 2 in the case of finite variances) and γ is the strong Diophantine type of α. This shows a change of behavior of the discrepancy at βγ = 2. In this paper we determine the exact order of magnitude of DN(Skα) for βγ < 1, and determine the limit distribution of N−1/2DN(Skα). We also prove a functional version of these results describing the asymptotic behavior of a wide class of functionals of the sequence {Skα}. Finally, we extend our results to the discrepancy of {Sk} for general random walks Sk without arithmetic conditions on X1, assuming only a mild polynomial rate on the weak convergence of {Sk} to the uniform distribution.

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