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2024 | OriginalPaper | Chapter

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

Authors : Jasmin Fiedler, Michael Gnewuch, Christian Weiß

Published in: Monte Carlo and Quasi-Monte Carlo Methods

Publisher: Springer International Publishing

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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

$$ D_N^{*s} \le \sqrt{\alpha } \sqrt{A+B\frac{\ln (\log _2(N))}{s}}\sqrt{\frac{s}{N}} $$

with $\alpha = \zeta ^{-1}(2) \approx 1.73, A=1165$ 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 $A_s<A$. Second, we generalize the result to the case of the (extreme) discrepancy. The paper is complemented by a section where we show numerical results for the dependence of the parameter $A_s$ on s.

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previous chapter Numerical Computation of Risk Functionals in PDMP Risk Models

next chapter Unbiased Likelihood Estimation of Wright–Fisher Diffusion Processes

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Title: New Bounds for the Extreme and the Star Discrepancy of Double-Infinite Matrices
Authors: Jasmin Fiedler
Michael Gnewuch
Christian Weiß
Publisher: Springer International Publishing
Book: Monte Carlo and Quasi-Monte Carlo Methods
Print ISBN: 978-3-031-59761-9

Electronic ISBN: 978-3-031-59762-6

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-59762-6_11

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