Just a word of warning, the central limit theorem is a bit misleading. Assuming the sum independent and identically distributed random variables converges to a

Just a word of warning, the central limit theorem is a bit misleading. Assuming the sum independent and identically distributed random variables converges to a distribution, that distribution is not necessarily the Gaussian, but a larger family called the Levy-Stable distributions [1]. Levy-Stable distributions are "heavy-tailed" in that far away from zero they behave like (inverse) power laws. This is probably why you see so many power laws in nature (gravity, "small-world" networks, income distribution, galaxy distribution density, etc.). This was one of the central themes of Mandelbrot's soapbox, that power laws were more fundamental than normal distributions. Mandelbrot gets remembered for the highly symmetric and pretty fractal pictures but those images (Koch curve, Dragon Curve, Sierpinski gasket, etc.) are like a focusing on a sine wave when talking about Fourier analysis. The central limit theorem applies to sums of random variables with finite variance [2]. Once you relax the condition of finite variance, or finite mean for that matter, Levy-Stable distributions are the more likely result. [1] https://en.wikipedia.org/wiki/Stable_distribution https://en.wikipedia.org/wiki/Stable_distribution [2] https://en.wikipedia.org/wiki/Central_limit_theorem https://en.wikipedia.org/wiki/Central_limit_theorem " ... Mathematically, if X ... is a random sample ... taken from a population with mean ... and finite variance ... the limiting form of the distribution ... is the standard normal distribution." (emphasis mine)