Low-discrepancy Sequence - The Formula of Hlawka-Zaremba

The Formula of Hlawka-Zaremba

Let . For we write

$dx_u:=\prod_{j\in u} dx_j$

and denote by the point obtained from x by replacing the coordinates not in u by . Then

$\frac{1}{N} \sum_{i=1}^N f(x_i) - \int_{\bar I^s} f(u)\,du= \sum_{\emptyset\neq u\subseteq D}(-1)^{|u|} \int_{^{|u|}}{\rm disc}(x_u,1)\frac{\partial^{|u|}}{\partial x_u}f(x_u,1) dx_u.$

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