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.

翻译：我们提出了一种灵活、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。该方法基于通过牛顿-柯特斯规则对线性卷积进行迭代数值积分。利用卷积密度函数的周期特性与梯形规则相结合，形成了一种稳健且高效的方法；该方法具有灵活性，可适用于各类非负连续随机变量。我们进行了误差分析，并研究了采用牛顿-柯特斯规则相较于快速傅里叶变换（FFT）进行数值积分的优势，结果表明：尽管使用FFT可能带来效率优势，但牛顿-柯特斯规则更能保持相对误差的稳定性，且其计算成本在可接受范围内。针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验证了我们的理论结果。