In statistical theory, convolutions are often avoided in favor of asymptotic approximation or simulation. Much of this is due to the fact that convolution is a challenging problem. With abundant computational resources, numerical convolution is a more viable option than in past decades. This paper proposes mathematical error bounds for the cumulative distribution function of the convolution of a finite number of independent univariate random variables. The discrete Fourier transform and its companion, the inverse discrete Fourier transform, are used to provide fast and easily obtainable mathematical error bounds for these convolutions. Examples and applications are provided to demonstrate a few possible uses of the error bounds.

中文翻译：

---

离散傅里叶变换的卷积累积分布函数的误差界

在统计理论中，通常避免使用卷积，而采用渐近逼近或模拟。这在很大程度上是由于卷积是一个具有挑战性的问题。拥有丰富的计算资源，数值卷积是比过去几十年更可行的选择。本文提出了有限个独立单变量随机变量卷积的累积分布函数的数学误差界。离散傅立叶变换及其伴随的逆傅立叶逆变换用于为这些卷积提供快速且容易获得的数学误差范围。提供了示例和应用程序，以说明误差范围的几种可能用法。