Error Bounds for Cumulative Distribution Functions of Convolutions via the Discrete Fourier Transform

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.