# ## Heavy-Tail Phenomena: Probabilistic and Statistical Modeling

 Heavy-tail analysis is a branch of extreme-value theory devoted to studying phenomena governed by large movements rather than gradual ones. It encompasses both probability modeling as well as statistical inference. Its mathematical tools are based on regular variation, weak convergence of probability measures and random measures and point processes. Its applications are diverse, including the following: • data networks, where the presence of heavy-tailed file sizes on network servers leads to long range dependence in the traffic rates; • finance, where financial returns are heavy tailed and thus risk management calculations of value-at-risk require heavy-tailed methods; • insurance, where the field of reinsurance is, by its nature, obsessed with very large values. There is an introductory chapter to describe the flavor and applicability of the subject. Then there are two chapters termed crash courses: one on regular variation and the other on weak convergence. These chapters contain essential material that could have been relegated to appendices; however, you should go through them where they are placed in the book. If you know the material, move quickly. Otherwise, pay some attention to style and notation. In particular, note what goes on in Sections 3.4–3.6. Such chapters are, inevitably, a compromise between wanting the book to be self-contained and not wanting to duplicate at length what is standard in other excellent references. Chapter 4 gets you into the heart of inference issues fairly quickly. The approach to inference is semiparametric and asymptotic in nature. This leads to a statistical theory that is different from classical contexts. We assume there is some structure out there at asymptopia and we are trying to infer what it is using a pitiful finite sample whose true model has not yet converged to the asymptotic model. Thus, maximum likelihood methods are not really available unless we simply assume from some threshold onwards that the asymptotic model holds. We give some diagnostics that help decide on values of parameters and when a heavy-tail model is appropriate. Chapter 5 begins the probability treatment which is geared towards a dimensionless theory. It focuses on the Poisson process and stochastic processes derived from the Poisson process, including Lévy and extremal processes. We also give an introduction to data network modeling. Chapter 6 gives the dimensionless treatment of regular variation and its probabilistic equivalents. We survey weak convergence techniques and discuss why it is difficult to bootstrap heavy-tail phenomena. Chapter 7 exploits the weak convergence technology to discuss weak convergence of extremes to extremal processes and weak convergence of summation processes to Lévy limits. Special cases include sums of heavy-tailed iid random variables converging to α-stable Lévy motion. We close the chapter with a unit on how weak convergence techniques can be used to study various transformations of regularly varying random vectors. We include Tauberian theory for Laplace transforms in this discussion. Applied probability takes center stage in Chapter 8 which uses heavy-tail techniques to learn about the properties of three models. Two of the models are for data networks and the last one is a more traditional queueing model. We return to statistical issues in Chapter 9, discussing asymptotic normality for estimators and then moving to inference for multivariate heavy-tailed models. We include examples of analysis of exchange rate data, Internet data, telephone network data and insurance data. Finally, we close the chapter with a discussion of the much praised and vilified sample correlation function. There are some appendices devoted to notational conventions and a list of symbols and also a section which timidly discusses some useful software. Each chapter contains exercises. Ignoring the exercises guarantees voyeur status.

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