Weak convergence to the t-distribution

We present a new limit theorem for random means: if the sample size is not deterministic but has a negative binomial or geometric distribution, the limit distribution of the normalised random mean is a t-distribution with degrees of freedom depending on the shape parameter of the negative binomial distribution. Thus the limit distribution exhibits exhibits heavy tails, whereas limit laws for random sums do not achieve this unless the summands have in nite variance. The limit law may help explain several empirical regularities. We consider two such examples: rst, a simple model is used to explain why city size growth rates are approximately t-distributed. Second, a random averaging argument can account for the heavy tails of high-frequency returns. Our empirical investigations demonstrate that these predictions are borne out by the data.