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Densities and Local Probabilities

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An Introduction to Heavy-Tailed and Subexponential Distributions

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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.

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Authors and Affiliations

  1. Department of Actuarial Mathematics, Heriot-Watt University, Riccarton, Edinburgh, UK

    Sergey Foss & Stan Zachary

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Dmitry Korshunov

Authors
  1. Sergey Foss
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  2. Dmitry Korshunov
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  3. Stan Zachary
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Foss, S., Korshunov, D., Zachary, S. (2013). Densities and Local Probabilities. In: An Introduction to Heavy-Tailed and Subexponential Distributions. Springer Series in Operations Research and Financial Engineering. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7101-1_4

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