Bounds of random star discrepancy for HSFC-based sampling

This paper is dedicated to the estimation of the probabilistic upper bounds of star discrepancy for Hilbert's space filling curve (HSFC) sampling. The primary concept revolves around the stratified random sampling method, with the relaxation of the stringent requirement for a sampling number $N=m^d$ in jittered sampling. We leverage the benefits of this sampling method to achieve superior results compared to Monte Carlo (MC) sampling. We also provide applications of the main result, which pertain to weighted star discrepancy, $L_2$-discrepancy, integration approximation in certain function spaces and examples in finance.