Some non-standard approaches to the study of sums of heavy-tailed distributions

Abstract

Heavy-tailed phenomena arise whenever events with very low probability have sufficiently large consequences that these events cannot be treated as negligible. These are sometimes described as low intensity, high impact events. Sums of heavy-tailed random variables play a major role in many areas of applied probability, for instance in risk theory, insurance mathematics, financial mathematics, queueing theory, telecommunications and computing, to name but a few areas. The theory of the asymptotic behaviour of a sum of independent heavy-tailed random variables is well-understood. We give a review of known results in this area, stressing the importance of some insensitivity properties of the class of long-tailed distributions. We introduce the new concept of the Boundary Class for a long-tailed distribution, and describe some of its properties and uses. We give examples of calculating the boundary class. Geometric sums of random variables are a useful model in their own right, for instance in reliability theory, but are also useful because they model the maximum of a random walk, which is itself a model that occurs in many applications. When the summands are heavy-tailed and independent then the asymptotic behaviour has been known since the 1970s. The asymptotic expression for the geometric sum is often used as an approximation to the actual distribution, owing to the (usually) analytically intractable form of the exact distribution. However the accuracy of this asymptotic approximation can be very poor, as we demonstrate. Following and further developing work by Kalashnikov and Tsitsiashvili we construct an upper bound for the relative accuracy of this approximation. We then develop new techniques for the application of our analytical results, and apply these in practice to several examples. Source code viii for the computer algorithms used in these calculations is given. As we have said, the asymptotic behaviour of a sum of heavy-tailed random variables is well-understood when the random variables are independent, the main characteristic being the principle of the single big jump. However, the case when the random variables are dependent is much less clear. We study this case for both deterministic and random sums using a novel approach, by considering conditional independence structures on the random variables. We seek sufficient conditions for the results of the theory with independent random variables still to hold. We give several examples to show how to apply and check our conditions, and the examples demonstrate a variety of effects owing to the dependence, and are also interesting in their own right. All the results we develop on this topic are entirely new. Some of the examples also include results that are new and have not been obtainable through previously existing techniques. For some examples we study the asymptotic behaviour is known, and this allows us to contrast our approach with previous approaches.