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This is a survey of some of the mathematical, probabilistic and statistical tools used in heavy-tail analysis as well as some examples of their use. Heavy tails are characteristic of phenomena where the probability of a huge value is relatively big. Record-breaking insurance losses, financial log-returns, file sizes stored on a server, transmission rates of files are all examples of heavy-tailed phenomena. The modeling and statistics of such phenomena are tail dependent and much different than classical modeling and statistical analysis, which give primacy to central moments, averages, and the normal density, which has a wimpy, light tail.

The next two chapters are rapid overviews of two essential subjects: regular variation and weak convergence. This kind of material is sometimes relegated to appendices, which is an unloved practice requiring much paging forward and back. Readers who are familiar with these subjects will find these chapters reassuring collections of notation and basic results. Those readers with less familiarity should read through the chapters to gain some functionality with the topics without worrying about all details; they can return later to ponder details, get further references, and improve mastery as time and circumstances allow. Other treatments and more detail can be found in [26, 90, 102, 144, 260, 275].

Asymptotic properties of statistics in heavy-tailed analysis are clearly understood with an interpretation which comes from the modern theory of weak convergence of probability measures on metric spaces, as originally promoted in [22] and updated in [25]. Additionally, utilizing the power of weak convergence allows for a rather unified treatment of the one-dimensional and higher-dimensional cases of heavy-tailed phenomena.

This material is designed to give immediate payoff for the previous two chapters. We give some estimators of the tail index, prove consistency, and evaluate the effectiveness of the estimation. We will return to statistical inference problems on several occasions, and the present chapter is a first experience with the statistical side of the subject. In particular, we will return to issues of asymptotic normality of the estimators in Chapter 9.1.

There are many fascinating and useful connections between heavy tails and the Poisson process, some of which we begin to describe here. Many heavy-tailed models are constructed from Poisson processes, which are the most tractable models of point systems. Some of these contructions give paradigms in the theory and some are elegant abstractions of applied systems.

This chapter discusses the relationship between (multivariate) regular variation and the Poisson process. We begin with a survey of multivariate regular variation as it applies to distributions. The goal is to make the results of Theorem 3.6 applicable to higher dimensions.

This chapter exploits connections between regular variation and the Poisson process given in Theorems 6.2 (p. 179) and 6.3 (p. 180) to understand several limit theorems and also to understand how regular variation of distributions of random vectors is transmitted by various transformations on the vectors. The fundamental philosophy is that we should capitalize on the equivalence between the analytical concept of regular variation and the probabilistic notion of convergence of empirical measures to limiting Poisson random measures.

This chapter uses the heavy-tail machinery in service of various applied probability models of networks and queuing systems.

This chapter surveys some additional statistical topics and presents analysis of several data sets to illustrate the techniques. One focus is multivariate inference: We consider methods for estimating the limit measure and the angular measure . These methods require statistical techniques for transforming the multivariate data to the case. We also consider the and an elaborating concept called , which aid in considering models possessing asymptotic independence. Finally, we consider a standard time-series tool called the and discuss its properties in the case of a stationary time series with heavy-tailed marginal distribtions.

Vectors are denoted by bold letters, uppercase for random vectors and lowercase for nonrandom vectors. For example, = (, ...., ) ∈ ℝ. Operations between vectors should be interpreted componentwise, so that for two vectors and , and so on. Also, define where in and , the “1” occurs in the th spot. For a real number , write = (, ..., ), as usual. We denote the rectangles (or the higher-dimensional intervals) by Higher-dimensional rectangles with one or both endpoints open are defined analogously, for example,