ON THE UPPER BOUND OF THE Lp DISCREPANCY OF HALTON’S SEQUENCE AND THE CENTRAL LIMIT THEOREM FOR HAMMERSLEY’S NET, II

Mordechay B. Levin

doi:10.1007/s10958-024-07312-9

ON THE UPPER BOUND OF THE Lp DISCREPANCY OF HALTON’S SEQUENCE AND THE CENTRAL LIMIT THEOREM FOR HAMMERSLEY’S NET, II

Mordechay B. Levin

Department of Mathematics

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

Abstract

Let H_s+1,N=(Hs(n),n/N)_n=0^N-1 be the s+1-dimensional Hammersley’s point set. Let D(x¯,(Hs+1,N)_n=0^N-1) be the local discrepancy of (Hs+1,N)_n=0^N-1, and let D_s,p((Hs+1,N)_n=0^N-1) be the L_p discrepancy of (Hs+1,N)_n=0^N-1. In this Part II of our paper, we prove the Central Limit Theorem for Hammersley’s net: (Formula presented.) where x¯ is a uniformly distributed random variable in [0,1]^s+1 and we claim that the lower bound of L_p discrepancy for p>0 is of the same order as the upper bound: (Formula presented.)

Original language	English
Pages (from-to)	1030-1064
Number of pages	35
Journal	Journal of Mathematical Sciences (United States)
Volume	280
Issue number	6
DOIs	https://doi.org/10.1007/s10958-024-07312-9
State	Published - Apr 2024

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.

Keywords

Central limit theorem
Halton’s sequence

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10.1007/s10958-024-07312-9

Cite this

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title = "ON THE UPPER BOUND OF THE Lp DISCREPANCY OF HALTON{\textquoteright}S SEQUENCE AND THE CENTRAL LIMIT THEOREM FOR HAMMERSLEY{\textquoteright}S NET, II",

abstract = "Let Hs+1,N=(Hs(n),n/N)n=0N-1 be the s+1-dimensional Hammersley{\textquoteright}s point set. Let D(x¯,(Hs+1,N)n=0N-1) be the local discrepancy of (Hs+1,N)n=0N-1, and let Ds,p((Hs+1,N)n=0N-1) be the Lp discrepancy of (Hs+1,N)n=0N-1. In this Part II of our paper, we prove the Central Limit Theorem for Hammersley{\textquoteright}s net: (Formula presented.) where x¯ is a uniformly distributed random variable in [0,1]s+1 and we claim that the lower bound of Lp discrepancy for p>0 is of the same order as the upper bound: (Formula presented.)",

keywords = "Central limit theorem, Halton{\textquoteright}s sequence",

author = "Levin, \{Mordechay B.\}",

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year = "2024",

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doi = "10.1007/s10958-024-07312-9",

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N2 - Let Hs+1,N=(Hs(n),n/N)n=0N-1 be the s+1-dimensional Hammersley’s point set. Let D(x¯,(Hs+1,N)n=0N-1) be the local discrepancy of (Hs+1,N)n=0N-1, and let Ds,p((Hs+1,N)n=0N-1) be the Lp discrepancy of (Hs+1,N)n=0N-1. In this Part II of our paper, we prove the Central Limit Theorem for Hammersley’s net: (Formula presented.) where x¯ is a uniformly distributed random variable in [0,1]s+1 and we claim that the lower bound of Lp discrepancy for p>0 is of the same order as the upper bound: (Formula presented.)

AB - Let Hs+1,N=(Hs(n),n/N)n=0N-1 be the s+1-dimensional Hammersley’s point set. Let D(x¯,(Hs+1,N)n=0N-1) be the local discrepancy of (Hs+1,N)n=0N-1, and let Ds,p((Hs+1,N)n=0N-1) be the Lp discrepancy of (Hs+1,N)n=0N-1. In this Part II of our paper, we prove the Central Limit Theorem for Hammersley’s net: (Formula presented.) where x¯ is a uniformly distributed random variable in [0,1]s+1 and we claim that the lower bound of Lp discrepancy for p>0 is of the same order as the upper bound: (Formula presented.)

KW - Central limit theorem

KW - Halton’s sequence

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ER -

ON THE UPPER BOUND OF THE Lp DISCREPANCY OF HALTON’S SEQUENCE AND THE CENTRAL LIMIT THEOREM FOR HAMMERSLEY’S NET, II

Abstract

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Keywords

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