缺点集的概率星离散界
Probabilistic Star Discrepancy Bounds for Lacunary Point Sets
摘要：
根据Heinrich，Novak，Wasilkowski和Woonniakowski的结果，星差$n(d,\varepsilon)$的逆满足$n(d,\varepsilon)\leq c_{\abs}d\varepsilon^{-2}$ .等价地，对于任何$N$和$d$，存在$[0,1)^d$中的一组$N$点，其星离散以$\sqrt{c_{\abs}\cdot d/N}$为界。他们实际上证明了一组独立的均匀分布的随机点以正概率满足这个上界。虽然Aistleitner和Hessel后来通过证明$c_{\abs}$的精确值来改进这个结果，这取决于不等式成立的概率，但到目前为止还没有这样一组已知点的一般构造。本文考虑了序列$(x_n)_{n\geq 1}=(\langle 2^{n-1}x_1\rangle)_{n\geq 1}$对一个均匀分布点$x_1\in [0,1)^d$的影响，证明了星差是由$C\sqrt{d\log_2d/N}$有界的. $C$的精确值取决于这个上界成立的概率。
英文摘要：
By a result of Heinrich, Novak, Wasilkowski and Woźniakowski the inverse of the star discrepancy $n(d,\varepsilon)$ satisfies $n(d,\varepsilon)\leq c_{\abs}d\varepsilon^{-2}$. Equivalently for any $N$ and $d$ there exists a set of $N$ points in $[0,1)^d$ with star discrepacny bounded by $\sqrt{c_{\abs}\cdot d/N}$. They actually proved that a set of independent uniformly distributed random points satisfies this upper bound with positive probability. Although Aistleitner and Hofer later refined this result by proving a precise value of $c_{\abs}$ depending on the probability with which the inequality holds, so far there is no general construction for such a set of points known. In this paper we consider the sequence $(x_n)_{n\geq 1}=(\langle 2^{n-1}x_1\rangle)_{n\geq 1}$ for a uniformly distributed point $x_1\in [0,1)^d$ and prove that the star discrepancy is bounded by $C\sqrt{d\log_2d/N}$. The precise value of $C$ depends on the probability with which this upper bound holds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
