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Mathematics of Computation

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Statistical properties of $b$-adic diaphonies

Author: Raffaello Seri
Journal: Math. Comp. 86 (2017), 799-828
MSC (2010): Primary 65D30; Secondary 62E20, 62E17
Published electronically: October 20, 2016
MathSciNet review: 3584549
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Abstract: 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

Keywords: $b$-adic diaphony, equidistribution, approximation of distributions, quadratic forms in Gaussian random variables
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
Article copyright: © Copyright 2016 American Mathematical Society