2020/06/11 by Jaspar Wiart, Elaine Wong · 7 citations
Computer Science · Mathematics · #Applied mathematics #Base (topology) #Covariance #Discrete mathematics #Estimator #Function (biology) #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Mathematical analysis #Mathematics #Numerical Methods and Algorithms #Statistics #cs.NA #cs.SC #math.NA
paper · pdf · doi:10.1016/j.matcom.2020.10.026
published in Mathematics and Computers in Simulation 182, 277-295 (Elsevier BV) · 27 pages; Supplementary material at https://wongey.github.io/digital-nets-walsh/
arxiv created 2020/06/11 · openalex created_date 2020/06/19 · openalex publication_date 2020/11/10 · arxiv updated 2020/11/20 · openalex updated_date 2026/08/05
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.