独立随机变量和左尾的快速精确数值方法
A Fast and Accurate Numerical Method for the Left Tail of Sums of Independent Random Variables
AI总结：
本文提出一种基于牛顿-柯特斯规则迭代卷积的确定性数值方法，用于高效计算非负独立随机变量和的左尾稀有事件概率，并通过误差分析和数值实验验证其相对误差保持优于FFT。
摘要：
我们提出一种灵活、确定性的数值方法，用于计算非负独立随机变量和的左尾稀有事件。该方法基于利用牛顿-柯特斯规则对线性卷积进行迭代数值积分。卷积密度的周期性性质与梯形规则相结合，被用来产生一种稳健且高效的方法，并且该方法具有灵活性，可应用于所有类型的非负连续随机变量。我们给出了误差分析，并研究了利用牛顿-柯特斯规则相对于快速傅里叶变换（FFT）进行数值积分的优势，表明尽管使用FFT可能带来效率上的好处，但牛顿-柯特斯规则倾向于更好地保持相对误差，并且确实以可接受的计算代价做到了这一点。对已知和未知稀有事件概率问题的数值研究展示了该方法的性能，并支持了我们的理论发现。
英文摘要：
We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons-Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton-Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency-benefits to using FFT, Newton-Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method's performance and support our theoretical findings.
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