Law of Large Numbers, Heavy-Tailed Distributions, and the Recent Financial Crisis

Introduction

In this paper we examine some possible inter-relationships between, on the one hand, the law of large numbers as it applies to cumulative sums of independent and identically distributed random variables where the distribution of each random variable is heavy-tailed, and on the other hand, the recent financial crisis. For the purposes of the present article, a distribution function ${\it \Phi}_{X}(\cdot)$ of a random variable X is said to be ’heavy-tailed’ if X has a finite first moment (i.e. an expected value) but not a finite second moment. It is of course possible to construct distribution functions where even the first moment fails to exist. But this case is of no interest to us, because in such a situation it would not make sense to talk about ’law of large numbers’.