Tractability properties of the weighted star discrepancy of the Halton sequence
This provides the weakest known condition for tractability of weighted star discrepancy for explicit low-discrepancy sequences, advancing theoretical understanding of quasi-Monte Carlo methods.
The paper proves that the Halton sequence achieves strong polynomial tractability for weighted star discrepancy under the mildest known condition on product weights, requiring only that the supremum over all subsets of products of indices and weights is finite. The same result extends to Niederreiter and other digital sequences, as well as to weighted unanchored discrepancy.
We study the weighted star discrepancy of the Halton sequence. In particular, we show that the Halton sequence achieves strong polynomial tractability for the weighted star discrepancy for product weights $(γ_j)_{j \ge 1}$ under the mildest condition on the weight sequence known so far for explicitly constructive sequences. The condition requires $\sup_{d \ge 1} \max_{\emptyset \not= \mathfrak{u} \subseteq [d]} \prod_{j \in \mathfrak{u}} (j γ_j) < \infty$. The same result holds for Niederreiter sequences and for other types of digital sequences. Our results are true also for the weighted unanchored discrepancy.