Star discrepancy bounds based on Hilbert space filling curve stratified sampling and its applications

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.