The central limit theorems all say that if you convolve stuff enough, and that stuff is sufficiently nice, the result will be a Gaussian distribution. How much is enough, and how nice is sufficient? Identically-distributed distributions converge quickly For many distributions d, the repeated convolution d∗d∗⋯∗d looks Gaussian. The number of convolutions you need to look Gaussian depends on the shape of d. This is the easiest variant of the central limit theorem: identically-distributed distributions. The uniform distribution converges real quick: The result of uniform(1, 2) * uniform(1, 2) * ... * uniform(1, 2), with 30 distributions total. This plot is an animated version of the plots in the previous post. The black curve is the Gaussian distribution with the same mean and variance as the red distribution. The more similar red is to black, the more Gaussian the result of the convolutions is. The numbers on the x axis are increasing because the mean of f∗g is the sum of the means of f and g, so if we start with positive means, repeated convolutions shoot off into higher numbers. Similar for the variance - notice how the width starts as the difference between 1 and 2, but ends with differences in the tens. You can keep the location stationary under convolution by starting with a distribution centered at 0, but you can't keep the variance from increasing, because you can't have a variance of 0 (except in the limiting case). Here's a more skewed distribution: beta(50, 1). beta(50, 1) is the probability distribution that represents knowing that a lake has bass and carp, but not how many of each, and then catching 49 bass in a row. It's fairly skewed! This time, after 30 convolutions, we're not quite Gaussian - the skew is still hanging around. But for a lot of real applications, I'd call the result "Gaussian enough". beta(50, 1) convolved with itself 30 times. A similar skew in the opposite direction, from the exponential distribution: exp(20) I was surprise