Central Limit Theorem for the Continuous Uniform Distribution

If is the PDF of a random variable , then , , (the convolution of and ). This means that finding the PDF of involves computing the -fold convolution of with itself, a computationally intensive operation to do directly even for small and simple PDF functions. This can be made easier in some cases by using the convolution property of Fourier transforms, , and the observation that the characteristic function of is the same as the Fourier transform , which means that can be computed by rescaling the inverse Fourier Transform of . For the uniform distribution, this lets us compute the distribution for much more quickly than we could directly, though it is still a little slow for larger .