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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Statistical properties of $b$-adic diaphonies
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by Raffaello Seri PDF
Math. Comp. 86 (2017), 799-828 Request permission


The aim of this paper is to derive the asymptotic statistical properties of a class of discrepancies on the unit hypercube called $b$-adic diaphonies. They have been introduced to evaluate the equidistribution of quasi-Monte Carlo sequences on the unit hypercube. We consider their properties when applied to a sample of independent and uniformly distributed random points. We show that the limiting distribution of the statistic is an infinite weighted sum of chi-squared random variables, whose weights can be explicitly characterized and computed. We also describe the rate of convergence of the finite-sample distribution to the asymptotic one and show that this is much faster than in the classical Berry-Esséen bound. Then, we consider in detail the approximation of the asymptotic distribution through two truncations of the original infinite weighted sum, and we provide explicit and tight bounds for the truncation error. Numerical results illustrate the findings of the paper, and an empirical example shows the relevance of the results in applications.
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Additional Information
  • Raffaello Seri
  • Affiliation: Dipartimento di Economia, Università degli Studi dell’Insubria, Via Monte Generoso 71, 21100 Varese, Italy – and – Center for Nonlinear and Complex Systems, Università degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy
  • MR Author ID: 710036
  • Email:
  • Received by editor(s): October 13, 2014
  • Received by editor(s) in revised form: August 16, 2015
  • Published electronically: October 20, 2016
  • Additional Notes: The author thanks Kiwi Cave Rafting, Waitomo Caves, New Zealand, for permission to use one of their pictures of the glowworm caves. Financial support through the COFIN Project 2010J3LZEN_006 is gratefully acknowledged. The usual disclaimer applies
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 799-828
  • MSC (2010): Primary 65D30; Secondary 62E20, 62E17
  • DOI:
  • MathSciNet review: 3584549