On numerical computation for the distribution of the convolution of N independent rectified Gaussian variables

Abstract

For large N and when no variables is predominant over the others, the central limit theorem (CLT) shall apply to the sum of random variables with negative values reset to zero. The parameters of the normal distribution are simply obtained by computing the expected value and the variance of each left rectified distributions. But for small N, the distribution of the sum is clearly not Gaussian and can present several modes and a strong skewness. In this paper, a way of computing the probability density function of the sum of N independent rectified Gaussian variables is presented, so that the calculation issues raised by the convolution product is solved. Some numerical examples are given and the validity of this approach is assessed through a comparison with a Monte-Carlo approach and an application to the PAH’s (Polycyclic Aromatic Hydrocarbon) batch filters measurements is provided.