Walsh functions, scrambled (0,m,s)-nets, and negative covariance: Applying symbolic computation to quasi-Monte Carlo integration

Jaspar Wiart and Elaine Wong

Mathematics and Computers in Simulation (MATCOM), 2021, vol. 182, issue C, 277-295

Abstract: We investigate base b Walsh functions for which the variance of the integral estimator based on a scrambled (0,m,s)-net in base b is less than or equal to that of the Monte-Carlo estimator based on the same number of points. First we compute the Walsh decomposition for the joint probability density function of two distinct points randomly chosen from a scrambled (t,m,s)-net in base b in terms of certain counting numbers and simplify it in the special case t is zero. Using this, we obtain an expression for the covariance of the integral estimator in terms of the Walsh coefficients of the function. Finally, we prove that the covariance of the integral estimator is negative when the Walsh coefficients of the function satisfy a certain decay condition. To do this, we use creative telescoping and recurrence solving algorithms from symbolic computation to find a sign equivalent closed form expression for the covariance term.

Keywords: Quasi-Monte Carlo integration; Scrambled digital nets; Walsh functions; Symbolic computation; Creative telescoping; Symbolic summation (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:182:y:2021:i:c:p:277-295

DOI: 10.1016/j.matcom.2020.10.026

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