New Bounds for the Extreme and the Star Discrepancy of Double-Infinite Matrices
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908755805863936 |
|---|---|
| author | Fiedler, Jasmin Gnewuch, Michael Weiß, Christian |
| author_facet | Fiedler, Jasmin Gnewuch, Michael Weiß, Christian |
| contents | 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 $$D_N^{*s} \leq \sqrtα \sqrt{A+B\frac{\ln(\log_2(N))}{s}}\sqrt{\frac{s}{N}}$$ with $α= ζ^{-1}(2) \approx 1.73, A=1165$ 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 $s$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_04686 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | New Bounds for the Extreme and the Star Discrepancy of Double-Infinite Matrices Fiedler, Jasmin Gnewuch, Michael Weiß, Christian Number Theory Numerical Analysis 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 $$D_N^{*s} \leq \sqrtα \sqrt{A+B\frac{\ln(\log_2(N))}{s}}\sqrt{\frac{s}{N}}$$ with $α= ζ^{-1}(2) \approx 1.73, A=1165$ 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 $s$. |
| title | New Bounds for the Extreme and the Star Discrepancy of Double-Infinite Matrices |
| topic | Number Theory Numerical Analysis |
| url | https://arxiv.org/abs/2305.04686 |