In this paper, we consider the upper bound of the probabilistic star discrepancy based on Hilbert space filling curve sampling. This problem originates from the multivariate integral approximation, but the main result removes the strict conditions on the sampling number of the classical grid-based jittered sampling. The main content has three parts. First, we inherit the advantages of this new sampling and achieve a better upper bound of the random star discrepancy than the use of Monte Carlo sampling. In addition, the convergence order of the upper bound is improved from $O(N^{-\frac{1}{2}})$ to $O(N^{-\frac{1}{2}-\frac{1}{2d}})$. Second, a better uniform integral approximation error bound of the function in the weighted space is obtained. Third, other applications will be given. Such as the sampling theorem in Hilbert spaces and the improvement of the classical Koksma-Hlawka inequality. Finally, the idea can also be applied to the proof of the strong partition principle of the star discrepancy version.

翻译：在本文中，我们考虑基于Hilbert空间曲线抽样的概率星偏差的上界。该问题源于多元积分逼近，但主要结果消除了对经典网格抖动抽样的采样数量的严格条件。本文的主要内容分为三个部分。首先，我们继承了这个新抽样的优点，并获得了比使用Monte Carlo抽样更好的随机星偏差上限。此外，上限的收敛阶数从$O(N^{-\frac{1}{2}})$改进为$O(N^{-\frac{1}{2}-\frac{1}{2d}})$。其次，在加权空间中获得了更好的函数均匀积分逼近误差界。第三，给出其他应用。例如，Hilbert空间的抽样定理和改进经典Koksma-Hlawka不等式。最后，这个想法还可以应用于星偏差版本的强分割原理的证明。