Is there a central limit theorem for power laws?

Abstract

Power laws can be found in various fields including physics, where its emergence is frequently associated with critical phenomena. Why is the power law so common? Perhaps like the normal distribution, the power law distribution also has its own central limit. In this FYP, we first verify the central limit theorem and the generalized central limit theorem for stable distributions through numerical methods. We then investigate the sum of power law random variables by sampling from the Pareto distribution and analyzing how the tail exponent of the resulting sum of power laws is affected. We also investigate the convergence of the sum of power laws to the alpha-stable distribution and for which values of α it converges to. Methods of generating power laws are also examined, along with the conditions when it stops generating power laws. We first start by using the uniform distribution to sample values as the parameter of an exponential distribution and investigate the resulting power law by varying the interval of values the uniform distribution can take. We then investigate another method using sums of lognormal random variables, where the number of random variables summed is determined by a binomial random variable and vary σ of the lognormal random variable to control the power law tail of the resulting distribution