An introduction to heavy-tailed and subexponential distributions. 1st ed.

*(English)*Zbl 1250.62025
Springer Series in Operations Research and Financial Engineering. New York, NY: Springer (ISBN 978-1-4419-9472-1/hbk; 978-1-4419-9473-8/ebook). ix, 123 p. (2011).

Heavy-tailed and subexponential distributions and their convolutions are the main theme of this monograph. There is wide area of applications of the theory of these distributions, e.g., in risk processes in finance, in the insurance-premia praxis, in epidemiological studies, etc. The characterisation and solving of the problems begins with considerations of heavy-tailed distributions all of whose positive exponential moments are infinite. The main results are lower limits for the convolution tails. The analysis of convolutions is based on a simple decomposition for such convolutions. The concept of \(h\)-insensitive distributions gives a technique for classifying long-tailed distributions and for the analysis of various classes of these distributions. All heavy-tailed distributions, likely to be of use in applications, posses the additional property of subexponentiality. The properties of subexponential distributions and structures of their classes are studied. The concept of the local subexponentiality gives insights into the local asymptotic behaviour of sums and maxima of random variables having heavy-tailed distributions. The last chapter is devoted to the study of random walks, whose increments have (a right) heavy-tailed distribution with a negative mean.

Reviewer: Pavel Froněk (Praha)

##### MSC:

62G32 | Statistics of extreme values; tail inference |

62E10 | Characterization and structure theory of statistical distributions |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

60G50 | Sums of independent random variables; random walks |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |