2023 | Book

# Closure Properties for Heavy-Tailed and Related Distributions

## An Overview

Authors: Remigijus Leipus, Jonas Šiaulys, Dimitrios Konstantinides

Publisher: Springer Nature Switzerland

Book Series : SpringerBriefs in Statistics

2023 | Book

Authors: Remigijus Leipus, Jonas Šiaulys, Dimitrios Konstantinides

Publisher: Springer Nature Switzerland

Book Series : SpringerBriefs in Statistics

This book provides a compact and systematic overview of closure properties of heavy-tailed and related distributions, including closure under tail equivalence, convolution, finite mixing, maximum, minimum, convolution power and convolution roots, and product-convolution closure. It includes examples and counterexamples that give an insight into the theory and provides numerous references to technical details and proofs for a deeper study of the subject. The book will serve as a useful reference for graduate students, young researchers, and applied scientists.

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Abstract

Investigation of various classes of heavy-tailed distributions attracted intense attention from theoreticians and practitioners because of their use in finance and insurance, communication networks, physics, hydrology, etc. Heavy-tailed distributions, whose most popular subclass is a class of regularly varying distributions, are also standard in applied probability when describing claim sizes in insurance mathematics, service times in queueing theory, and lifetimes of particles in branching process theory. In this book, we study the closure property of heavy-tailed and related distribution classes, which usually states that assuming two or more distributions in some specific class, the result of the corresponding operation (e.g. sum-convolution, product-convolution, mixture) belongs to the same class of distributions. The description of closure properties of a given distribution class is not only an interesting mathematical problem. Using closure properties of a given distribution class, one can effectively construct the representatives of the class and understand the mechanisms causing heavy tails in real life.

Abstract

In this chapter, we define the distribution classes whose closure properties will be considered in the subsequent chapters: heavy-tailed distributions, regularly varying distributions, and their generalizations—consistently varying and dominatedly varying distributions. Next, we consider the long-tailed distributions and their extensions to exponential-like-tailed and generalized long-tailed distributions. Finally, we introduce the subexponential and related distributions: strong subexponential, convolution-equivalent, and generalized subexponential distributions. We concentrate on the properties, characterization, and examples of these classes. Inclusion properties between the defined classes are explained.

Abstract

We start this chapter with a motivating discussion on the use of convolution closure when evaluating the ruin probability in the classical risk process. In Sect. 3.3, we discuss the convolution closure properties in relation to the notion of max-sum equivalence. In further sections, we overview and discuss the closure properties of the heavy-tailed and related distributions, introduced in Chap. 2, under strong/weak tail-equivalence, convolution, finite mixing, maximum, and minimum. Together, we show how these closure properties can be extended to the convolution power and order statistics. The corresponding closure properties are followed by discussions, numerous examples, and counterexamples.

Abstract

Besides the convolution closure, it is often of interest to understand whether the attribution of a distribution F to the specific class of distributions is caused by the inclusion of \(F^{*n}\) to the same family. Such an implication is called a convolution-root closure. This chapter is devoted to the convolution-root closure properties for the distribution classes described in Chap. 2. We determine the classes which are closed under convolution roots and which are not.

Abstract

Products of random variables and related distribution problems appear in physics, engineering, number theory, and many probability and statistical problems, such as multivariate statistical modelling, asymptotic analysis of randomly weighted sums, etc. In financial time series, the multiplicative structures occur in modelling conditional heteroskedasticity as in GARCH or stochastic volatility models. In this chapter, we mainly are interested in the following questions: (1) when a given class of distributions is closed with respect product-convolution?; (2) for which distributions G the product-convolution \(F \otimes G\) remains in the same class as a primary distribution F? (3) for given classes of distributions F and G, which class of distributions the product-convolution \(F \otimes G\) belongs to. Also, we discuss the phenomena of producing heavy tails from the light-tailed multipliers.

Abstract

This concluding chapter collects the closure properties for the heavy-tailed and related distribution classes, considered in the book. In order to see the whole picture for the validity of closure properties among the classes and compare them between themselves, we place them in Table 6.1.

Table 6.1

Closure properties for heavy-tailed and related distributions

Closure under: | \(\mathcal {R}(\alpha )\) | \(\mathcal {C}\) | \(\mathcal {L}\cap \mathcal {D}\) | \(\mathcal {D}\) | \(\mathcal {S}^*\) | \(\mathcal {S}\) | \(\mathcal {S}(\gamma )\) | \(\mathcal {O}\mathcal {S}\) | \(\mathcal {L}\) | \(\mathcal {L}(\gamma )\) | \(\mathcal {O}\mathcal {L}\) | \(\mathcal {H}\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Strong | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) |

tail-equivalence | (8) | (10) | (14) | (11) | (21) | (17) | (23) | (25) | (13) | (15) | (16) | (5) |

Weak | no | no | no | \(\boldsymbol {\checkmark }\) | no | no | no | \(\boldsymbol {\checkmark }\) | no | no | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) |

tail-equivalence | (9) | (10) | (14) | (11) | (21) | (19) | (24) | (25) | (14) | (15) | (16) | (5) |

Convolution | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | no | no | no | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) |

(8) | (10) | (14) | (11) | (2) | (19) | (24) | (25) | (13) | (15) | (16) | (5) | |

Convolution power | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) |

(9) | (11) | (14) | (13) | (22) | (25) | (25) | (26) | (14) | (16) | (16) | (7) | |

Convolution root | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | no | no | no | no | no | \(\boldsymbol {\checkmark }\) |

(1) | (1) | (1) | (1) | (1) | (1) | (2) | (2) | (2) | (2) | (2) | (1) | |

Product-convolution | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | no | \(\boldsymbol {\checkmark }\) | no | \(\boldsymbol {\checkmark } \) | \(\boldsymbol {\checkmark }\) | ||

(7) | (8) | (10) | (9) | (14) | (14) | (9) | (10) | (11) | (6) | |||

Mixture | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | no | no | no | no | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) |

(8) | (10) | (14) | (11) | (21) | (19) | (24) | (25) | (13) | (15) | (16) | (5) | |

Maximum | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | no | no | no | no | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) |

(8) | (10) | (14) | (11) | (21) | (19) | (24) | (26) | (13) | (15) | (16) | (5) | |

Minimum | \(\boldsymbol {\checkmark }^*\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }^{**}\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }\) | \(\boldsymbol {\checkmark }^{**}\) | \(\boldsymbol {\checkmark }\) | no |

(8) | (10) | (14) | (11) | (21) | (17) | (23) | (25) | (13) | (15) | (16) | (6) |