Randomized quasi-Monte Carlo methods have been introduced with the main purpose of yielding a computable measure of error for quasi-Monte Carlo approximations through the implicit application of a central limit theorem over independent randomizations. But to increase precision for a given computational budget, the number of independent randomizations is usually set to a small value so that a large number of points are used from each randomized low-discrepancy sequence to benefit from the fast convergence rate of quasi-Monte Carlo. While a central limit theorem has been previously established for a specific but computationally expensive type of randomization, it is also known in general that fixing the number of randomizations and increasing the length of the sequence used for quasi-Monte Carlo can lead to a non-Gaussian limiting distribution. This paper presents sufficient conditions on the relative growth rates of the number of randomizations and the quasi-Monte Carlo sequence length to ensure a central limit theorem and also an asymptotically valid confidence interval. We obtain several results based on the Lindeberg condition for triangular arrays and expressed in terms of the regularity of the integrand and the convergence speed of the quasi-Monte Carlo method. We also analyze the resulting estimator’s convergence rate.

引入随机准蒙特卡罗方法的主要目的是通过在独立随机化上隐式应用中心极限定理来产生准蒙特卡罗近似的可计算误差度量。但为了提高给定计算预算的精度，独立随机化的数量通常设置为较小的值，以便从每个随机低差异序列中使用大量点，以受益于准蒙特卡罗的快速收敛速度。虽然之前已经为特定但计算成本昂贵的随机化类型建立了中心极限定理，但众所周知，固定随机化的数量并增加用于准蒙特卡洛的序列长度可能会导致非-高斯极限分布。本文提出了随机数相对增长率和准蒙特卡罗序列长度的充分条件，以确保中心极限定理和渐近有效的置信区间。我们基于三角阵的Lindeberg条件得到了几个结果，并用被积函数的正则性和准蒙特卡罗方法的收敛速度来表示。我们还分析了所得估计器的收敛速度。