The Fundamentals of Heavy Tails
Properties, Emergence, and Estimation

An accessible yet rigorous package of probabilistic and statistical tools for anyone who must understand or model extreme events.

Jayakrishnan Nair (Author), Adam Wierman (Author), Bert Zwart (Author)

9781316511732, Cambridge University Press

Hardback, published 9 June 2022

264 pages
26 x 18.3 x 1.2 cm, 0.68 kg

'The book provides a fresh look at heavy-tailed probability distributions on the real line and their role in applied probability. The authors show that these distributions appear via natural algebraic operations. Their approach, towards understanding properties of these distributions, combines the key mathematical ideas alongside with informal explanations. Physical intuition is also provided, for example, the 'catastrophe/big jump principle' for heavy-tailed distributions versus the 'conspiracy principle' for light-tailed ones. The book is designed to help the practitioner and includes many interesting examples and exercises that may help to the reader to adjust and enjoy its content.' Sergey Foss, Heriot-Watt University

Heavy tails –extreme events or values more common than expected –emerge everywhere: the economy, natural events, and social and information networks are just a few examples. Yet after decades of progress, they are still treated as mysterious, surprising, and even controversial, primarily because the necessary mathematical models and statistical methods are not widely known. This book, for the first time, provides a rigorous introduction to heavy-tailed distributions accessible to anyone who knows elementary probability. It tackles and tames the zoo of terminology for models and properties, demystifying topics such as the generalized central limit theorem and regular variation. It tracks the natural emergence of heavy-tailed distributions from a wide variety of general processes, building intuition. And it reveals the controversy surrounding heavy tails to be the result of flawed statistics, then equips readers to identify and estimate with confidence. Over 100 exercises complete this engaging package.

Commonly used notation
1. Introduction
Part I. Properties: 2. Scale invariance, power laws, and regular variation
3. Catastrophes, conspiracies, and subexponential distributions
4. Residual lives, hazard rates, and long tails
Part II. Emergence: 5. Additive processes
6. Multiplicative processes
7. Extremal processes
Part III. Estimation: 8. Estimating power-law distributions: Listen to the body
9. Estimating power-law tails: Let the tail do the talking
References
Index.

Subject Areas: Computer science [UY], Electrical engineering [THR], Probability & statistics [PBT], Finance [KFF], Econometrics [KCH], Information theory [GPF]