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.

中文翻译：

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Walsh 函数、加扰 $(0,m,s)$-nets 和负协方差：将符号计算应用于准蒙特卡罗积分

我们研究了基 $b$ Walsh 函数，其中基于 $b$ 中加扰 $(0,m,s)$-net 的积分估计量的方差小于或等于基于 Monte-Carlo 估计量的方差在相同数量的点上。首先，我们计算两个不同点的联合概率密度函数的 Walsh 分解，这两个不同点随机选择在基 $b$ 中的加扰 $(t,m,s)$-net 中，并根据某些计数数字进行简化，并在特殊情况下对其进行简化$t$ 为零。使用它，我们根据函数的 Walsh 系数获得积分估计量协方差的表达式。最后，我们证明了当函数的 Walsh 系数满足一定的衰减条件时，积分估计量的协方差为负。去做这个，