Halton Sequences Avoid the Origin

The n’th point of the Halton sequence in [0, 1]d is shown to have components whose product is larger than Cn−1 where C > 0 depends on d. This property makes the Halton sequence very well suited to quasi-Monte Carlo integration of some singular functions that become unbounded as the argument approaches the origin. The Halton sequence avoids a similarly shaped (though differently sized) region around every corner of the unit cube, making it suitable for functions with singularities at all corners. Convergence rates are established for quasi-Monte Carlo integration based on growth conditions of the integrand, and measures of how the sample points avoid the boundary. In some settings the error is O(n−1+ ) while in others the error diverges to infinity. Star discrepancy does not suffice to distinguish the cases.