Central Limit Theorem for (t,s)-sequences in base 2

Mordechay B. Levin

doi:10.1016/j.jco.2022.101699

Central Limit Theorem for (t,s)-sequences in base 2

Mordechay B. Levin

Department of Mathematics

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

2 Scopus citations

Abstract

Let (x_n)_n≥0 be a digital (t,s)-sequence in base 2, P_m=(x_n)_n=0^{2^m−1}, and let D(P_m,Y) be the local discrepancy of P_m. Let T⊕Y be the digital addition of T and Y, and let M_s,p(P_m)=(∫[0,1)^2s|D(P_m⊕T,Y)|^pdTdY)^1/p. In this paper, we prove that D(P_m⊕T,Y)/M_s,2(P_m) weakly converges to the standard Gaussian distribution for m→∞, where T,Y are uniformly distributed random variables in [0,1)^s.

Original language	English
Article number	101699
Journal	Journal of Complexity
Volume	75
DOIs	https://doi.org/10.1016/j.jco.2022.101699
State	Published - Apr 2023

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.

Funding

I am very grateful to the two referees for many corrections and suggestions which improved this paper. Parts of this work were started at the Workshop “Discrepancy Theory and Quasi-Monte Carlo methods” held at the Erwin Schrödinger Institute, September 25 - 29, 2017.

Keywords

(t,s)-sequence
Central limit theorem
Discrepancy

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10.1016/j.jco.2022.101699

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abstract = "Let (xn)n≥0 be a digital (t,s)-sequence in base 2, Pm=(xn)n=02m−1, and let D(Pm,Y) be the local discrepancy of Pm. Let T⊕Y be the digital addition of T and Y, and let Ms,p(Pm)=(∫[0,1)2s|D(Pm⊕T,Y)|pdTdY)1/p. In this paper, we prove that D(Pm⊕T,Y)/Ms,2(Pm) weakly converges to the standard Gaussian distribution for m→∞, where T,Y are uniformly distributed random variables in [0,1)s.",

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N2 - Let (xn)n≥0 be a digital (t,s)-sequence in base 2, Pm=(xn)n=02m−1, and let D(Pm,Y) be the local discrepancy of Pm. Let T⊕Y be the digital addition of T and Y, and let Ms,p(Pm)=(∫[0,1)2s|D(Pm⊕T,Y)|pdTdY)1/p. In this paper, we prove that D(Pm⊕T,Y)/Ms,2(Pm) weakly converges to the standard Gaussian distribution for m→∞, where T,Y are uniformly distributed random variables in [0,1)s.

AB - Let (xn)n≥0 be a digital (t,s)-sequence in base 2, Pm=(xn)n=02m−1, and let D(Pm,Y) be the local discrepancy of Pm. Let T⊕Y be the digital addition of T and Y, and let Ms,p(Pm)=(∫[0,1)2s|D(Pm⊕T,Y)|pdTdY)1/p. In this paper, we prove that D(Pm⊕T,Y)/Ms,2(Pm) weakly converges to the standard Gaussian distribution for m→∞, where T,Y are uniformly distributed random variables in [0,1)s.

KW - (t,s)-sequence

KW - Central limit theorem

KW - Discrepancy

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Central Limit Theorem for (t,s)-sequences in base 2

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