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37 results

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  • 7 - Extremal Processes

  • from Part II - Emergence
  • Jayakrishnan Nair, Indian Institute of Technology, Bombay, Adam Wierman, California Institute of Technology, Bert Zwart, Stichting Centrum voor Wiskunde en Informatica (CWI), Amsterdam
  • Book:
    The Fundamentals of Heavy Tails
    Published online:
    17 May 2022
    Print publication:
    09 June 2022, pp 148-174

  • Summary

    An introduction to the emergence of heavy-tailed distributions in the context of extremal processes.Max-stable distributions are introduced, and the extremal central limit theory is presented.Further, an example of the emergence of heavy tails in the extremes of random walks is presented.

  • 1 - Introduction

  • Jayakrishnan Nair, Indian Institute of Technology, Bombay, Adam Wierman, California Institute of Technology, Bert Zwart, Stichting Centrum voor Wiskunde en Informatica (CWI), Amsterdam
  • Book:
    The Fundamentals of Heavy Tails
    Published online:
    17 May 2022
    Print publication:
    09 June 2022, pp 1-26
    • Chapter
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  • Summary

    An introduction to the class of heavy-tailed distributions, including formal definitions, basic properties, and examples of common distributions that are heavy-tailed.

  • 6 - Multiplicative Processes

  • from Part II - Emergence
  • Jayakrishnan Nair, Indian Institute of Technology, Bombay, Adam Wierman, California Institute of Technology, Bert Zwart, Stichting Centrum voor Wiskunde en Informatica (CWI), Amsterdam
  • Book:
    The Fundamentals of Heavy Tails
    Published online:
    17 May 2022
    Print publication:
    09 June 2022, pp 127-147

  • Summary

    An introduction to the emergence of heavy-tailed distributions in the context of multiplicative processes.The multiplicative central limit theorem is presented, and variations of multiplicative processes with lower barriers and noise are studied.Further, an example of the emergence of heavy tails in random graphs via preferential attachment is included.

  • 5 - Additive Processes

  • from Part II - Emergence
  • Jayakrishnan Nair, Indian Institute of Technology, Bombay, Adam Wierman, California Institute of Technology, Bert Zwart, Stichting Centrum voor Wiskunde en Informatica (CWI), Amsterdam
  • Book:
    The Fundamentals of Heavy Tails
    Published online:
    17 May 2022
    Print publication:
    09 June 2022, pp 107-126

  • Summary

    An introduction to the emergence of heavy-tailed distributions in the context of additive processes.Stable distributions are introduced and the generalized central limit theorem is presented.Further, an example of the emergence of heavy tails in random walks is included.

The Fundamentals of Heavy Tails
  • Properties, Emergence, and Estimation
  • Jayakrishnan Nair, Adam Wierman, Bert Zwart
  • Published online:
    17 May 2022
    Print publication:
    09 June 2022
  • 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.

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