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We investigate base $b$ Walsh functions for which the variance of theintegral estimator based on a scrambled $(0,m,s)$-net in base $b$ is less thanor equal to that of the Monte-Carlo estimator based on the same number ofpoints. First we compute the Walsh decomposition for the joint probabilitydensity 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 itin the special case $t$ is zero. Using this, we obtain an expression for thecovariance of the integral estimator in terms of the Walsh coefficients of thefunction. Finally, we prove that the covariance of the integral estimator isnegative when the Walsh coefficients of the function satisfy a certain decaycondition. To do this, we use creative telescoping and recurrence solvingalgorithms from symbolic computation to find a sign equivalent closed formexpression for the covariance term.

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