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2011 | Buch

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An Introduction to Heavy-Tailed and Subexponential Distributions

verfasst von: Sergey Foss, Dmitry Korshunov, Stan Zachary

Verlag: Springer New York

Buchreihe : Springer Series in Operations Research and Financial Engineering

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This monograph provides a complete and comprehensive introduction to the theory of long-tailed and subexponential distributions in one dimension. New results are presented in a simple, coherent and systematic way. All the standard properties of such convolutions are then obtained as easy consequences of these results. The book focuses on more theoretical aspects. A discussion of where the areas of applications currently stand in included as is some preliminary mathematical material. Mathematical modelers (for e.g. in finance and environmental science) and statisticians will find this book useful.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Heavy-tailed distributions (probability measures) play a major role in the analysis of many stochastic systems. For example, they are frequently necessary to accurately model inputs to computer and communications networks, they are an essential component of the description of many risk processes, they occur naturally in models of epidemiological spread, and there is much statistical evidence for their appropriateness in physics, geoscience and economics. Important examples are Pareto distributions (and other essentially power-law distributions), lognormal distributions, and Weibull distributions (with shape parameter less than 1). Indeed most heavy-tailed distributions used in practice belong to one of these families, which are defined, along with others, in Chap. 2. We also consider the Weibull distribution at the end of this chapter.

Sergey Foss, Dmitry Korshunov, Stan Zachary

Chapter 2. Heavy-Tailed and Long-Tailed Distributions

Abstract

In this chapter we are interested in (right-) tail properties of distributions, i.e. in properties of a distribution which, for any x, depend only on the restriction of the distribution to (x, ∞). More generally it is helpful to consider tail properties of functions.

Sergey Foss, Dmitry Korshunov, Stan Zachary

Chapter 3. Subexponential Distributions

Abstract

As we stated in the Introduction, all those heavy-tailed distributions likely to be of use in practical applications are not only long-tailed but possess the additional regularity property of subexponentiality. Essentially this corresponds to good tail behaviour under the operation of convolution. In this chapter, following established tradition, we introduce first subexponential distributions on the positive half-line \({\mathbb{R}}^{+}\). It is not immediately obvious from the definition, but it nevertheless turns out, that subexponentiality is a tail property of a distribution. It is thus both natural, and important for many applications, to extend the concept to distributions on the entire real line \(\mathbb{R}\). We also study the very useful subclass of subexponential distributions which was originally called \({\mathcal{S}}^{{_\ast}}\) in [29] and which we name strong subexponential. In particular this class again contains all those heavy-tailed distributions likely to be encountered in practice.

Sergey Foss, Dmitry Korshunov, Stan Zachary

Chapter 4. Densities and Local Probabilities

Abstract

This chapter is devoted to local long-tailedness and to local subexponentiality. First we consider densities with respect to either Lebesgue measure on \(\mathbb{R}\) or counting measure on \(\mathbb{Z}\). Next we study the asymptotic behaviour of the probabilities to belong to an interval of a fixed length. We give the analogues of the basic properties of the tail probabilities including two analogues of Kesten’s estimate, and provide sufficient conditions for probability distributions to have these local properties.

Sergey Foss, Dmitry Korshunov, Stan Zachary

Chapter 5. Maximum of Random Walk

Abstract

In this chapter, we study a random walk whose increments have a (right) heavy-tailed distribution with a negative mean. The maximum of such a random walk is almost surely finite, and our interest is in the tail asymptotics of the distribution of this maximum, for both infinite and finite time horizons; we are further interested in the local asymptotics for the maximum in the case of an infinite time horizon. We use direct probabilistic techniques and show that, under the appropriate subexponentiality conditions, the main reason for the maximum to be far away from zero is again that a single increment of the walk is similarly large.

Sergey Foss, Dmitry Korshunov, Stan Zachary

Backmatter

Titel: An Introduction to Heavy-Tailed and Subexponential Distributions
verfasst von: Sergey Foss
Dmitry Korshunov
Stan Zachary
Copyright-Jahr: 2011
Verlag: Springer New York
Electronic ISBN: 978-1-4419-9473-8
Print ISBN: 978-1-4419-9472-1
DOI: https://doi.org/10.1007/978-1-4419-9473-8