New Bounds for the Extreme and the Star Discrepancy of Double-Infinite Matrices

2023/05/08 by Jasmin Fiedler, Fiedler, Jasmin, Michael Gnewuch +3 · 2 citations
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT) #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2305.04686

openalex publication_date 2023/05/08 · openalex created_date 2023/05/10 · openalex updated_date 2026/07/28

Abstract

According to Aistleitner and Weimar, there exist two-dimensional (double) infinite matrices whose star-discrepancy D_N^*s of the first N rows and s columns, interpreted as N points in [0,1]^s, satisfies an inequality of the form D_N^*s ≤ √α √(A+B(ln(log₂(N)))/(s))√((s)/(N)) with α= ζ^-1(2) ≈ 1.73, A=1165 and B=178. These matrices are obtained by using i.i.d sequences, and the parameters s and N refer to the dimension and the sample size respectively. In this paper, we improve their result in two directions: First, we change the character of the equation so that the constant A gets replaced by a value A_s dependent on the dimension s such that for s>1 we have A_s

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• Sharp Non-Asymptotic Bounds for the Star Discrepancy of Double-Infinite Random Matrices via Optimal Covering Numbers
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