Upper bounds for generalized Lp-discrepancy of random points

We study the L_p-discrepancy of random point sets in high dimensions, with emphasis on small values of p. Although the classical L_p-discrepancy suffers from the curse of dimensionality for all p∈(1,∞), the gap between known upper and lower bounds remains substantial, in particular for small p≥1. To clarify this picture, we review the existing results for i.i.d. uniformly distributed points and derive new upper bounds for generalized L_p-discrepancies, obtained by allowing non-uniform sampling densities and corresponding non-negative quadrature weights. Using the probabilistic method, we show that random points drawn from optimally chosen product densities lead to significantly improved upper bounds. For p=2 these bounds are explicit and optimal; for general p∈[1,∞) we obtain sharp asymptotic estimates. The improvement can be interpreted as a form of importance sampling for the underlying Sobolev space F_d,q. Our results also reveal that, even with optimal densities, the curse of dimensionality persists for random points when p≥1, and it becomes most pronounced for small p. This suggests that the curse should also hold for the classical L₁-discrepancy for deterministic point sets.