Halton Sequences Avoid the Origin

The nth 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 (QMC) 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 QMC integration based on two assumptions: a growth condition on the integrand, and a measure of how the sample points avoid the boundary. In some settings the error is O(n-1 + epsilon ), while in others the error diverges to infinity. Star discrepancy does not suffice to distinguish the cases.