Weighted $L_p$-Discrepancy Bounds for Parametric Stratified Sampling and Applications to High-Dimensional Integration

This paper studies the expected $L_p$-discrepancy ($2\le p<\mathrm{\infty }$) for stratified sampling schemes under importance sampling. We introduce a parametric family of equivolume partitions ${\mathrm{\Omega }}_{\theta ,\sim }$ and leverage recent exact formulas for the expected $L_2$-discrepancy \cite{xian2025improved}. Our main contribution is a weighted discrepancy reduction lemma that relates weighted $L_p$-discrepancy to standard $L_p$-discrepancy with explicit constants depending on the weight function. For $p=2$, we obtain explicit bounds using the exact discrepancy formulas. For $p>2$, we derive probabilistic bounds via dyadic chaining techniques. The results yield uniform error estimates for multivariate integration in Sobolev spaces ${\mathcal{H}}^1(K)$ and $F_{d,q}^{\ast }$, demonstrating improved performance over classical jittered sampling in importance sampling scenarios. Numerical experiments validate our theoretical findings and illustrate the practical advantages of parametric stratified sampling.