How Exceptional is the Extremal Kendall and Kendall-Type Convolution
This paper deals with the generalized convolutions connected with the Williamson transform and the maximum operation. We focus on such convolutions which can define transition probabilities of renewal processes. They should be monotonic since the described time or destruction does not go back, it sh...
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| Published in: | Resultate der Mathematik Vol. 78; no. 6 |
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| Abstract | This paper deals with the generalized convolutions connected with the Williamson transform and the maximum operation. We focus on such convolutions which can define transition probabilities of renewal processes. They should be monotonic since the described time or destruction does not go back, it should admit existence of a distribution with a lack of memory property because the analog of the Poisson process shall exist. Another valuable property is the simplicity of calculating and inverting the corresponding generalized characteristic function (in particular Williamson transform) so that the technique of generalized characteristic function can be used in description of our processes. The convex linear combination property (the generalized convolution of two point measures is the convex combination of several fixed measures), or representability (which means that the generalized convolution can be easily written in the language of independent random variables)—they also facilitate the modeling of real processes in that language. We describe examples of generalized convolutions having the required properties ranging from the maximum convolution and its simplest generalization—the Kendall convolution (associated with the Williamson transform), up to the most complicated here—Kingman convolution. It is novel approach to apply in the extreme value theory. Stochastic representation of the Kucharczak-Urbanik in the order statistics terms is proved, which open new paths to investigate Archimedean copulas. This paper open the door to solve an old open problem of the relationship between copulas and generalized convolutions mentioned by B. Schweizer and A. Sklar in 1983. This indicates the path of further research towards extremes and dependency modelling. |
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| AbstractList | This paper deals with the generalized convolutions connected with the Williamson transform and the maximum operation. We focus on such convolutions which can define transition probabilities of renewal processes. They should be monotonic since the described time or destruction does not go back, it should admit existence of a distribution with a lack of memory property because the analog of the Poisson process shall exist. Another valuable property is the simplicity of calculating and inverting the corresponding generalized characteristic function (in particular Williamson transform) so that the technique of generalized characteristic function can be used in description of our processes. The convex linear combination property (the generalized convolution of two point measures is the convex combination of several fixed measures), or representability (which means that the generalized convolution can be easily written in the language of independent random variables)—they also facilitate the modeling of real processes in that language. We describe examples of generalized convolutions having the required properties ranging from the maximum convolution and its simplest generalization—the Kendall convolution (associated with the Williamson transform), up to the most complicated here—Kingman convolution. It is novel approach to apply in the extreme value theory. Stochastic representation of the Kucharczak-Urbanik in the order statistics terms is proved, which open new paths to investigate Archimedean copulas. This paper open the door to solve an old open problem of the relationship between copulas and generalized convolutions mentioned by B. Schweizer and A. Sklar in 1983. This indicates the path of further research towards extremes and dependency modelling. |
| ArticleNumber | 224 |
| Author | Wesołowski, Jacek Misiewicz, Jolanta K. Jasiulis-Gołdyn, Barbara H. Omey, Edward |
| Author_xml | – sequence: 1 givenname: Barbara H. orcidid: 0000-0003-2358-4494 surname: Jasiulis-Gołdyn fullname: Jasiulis-Gołdyn, Barbara H. email: barbara.jasiulis@math.uni.wroc.pl, basiaja@liverpool.ac.uk organization: Institute of Mathematics, University of Wrocław, Institute for Financial and Actuarial MathematicsDepartment of Mathematical Sciences, University of Liverpool – sequence: 2 givenname: Jolanta K. surname: Misiewicz fullname: Misiewicz, Jolanta K. organization: Faculty of Mathematics and Information Science, Warsaw University of Technology – sequence: 3 givenname: Edward surname: Omey fullname: Omey, Edward organization: Faculty of Economics and Business-Campus Brussels, KU Leuven – sequence: 4 givenname: Jacek surname: Wesołowski fullname: Wesołowski, Jacek organization: Faculty of Mathematics and Information Science, Warsaw University of Technology |
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| Cites_doi | 10.3150/17-BEJ977 10.4064/sm-45-1-57-70 10.1016/j.jmaa.2019.123575 10.1007/978-94-009-3859-5_11 10.1016/j.jmva.2010.03.015 10.1016/j.jmva.2019.01.007 10.1002/9781119157052.ch2 10.4064/cm-48-1-117-125 10.1007/s10959-005-7528-0 10.1215/S0012-7094-56-02317-1 10.4064/cm-25-2-281-289 10.4064/sm-23-3-217-245 10.3150/14-BEJ653 10.1007/s10986-015-9296-6 10.1007/s10959-008-0195-1 10.1016/j.spa.2019.09.013 10.1007/1-4020-4415-1 10.1002/0471722162 10.1007/s00025-023-01946-y 10.4064/sm167-3-1 10.1007/978-94-017-1758-8 10.1287/moor.2018.0950 10.1007/978-3-642-33483-2 10.1137/S0040585X97T987491 10.1007/s10986-017-9375-y 10.1007/s00010-018-0578-z 10.1007/s10959-010-0279-6 10.1111/j.1467-842X.2004.00330.x 10.1007/BF02391808 10.1007/s10463-022-00858-y 10.1214/07-AOS556 10.4064/sm-80-2-167-189 10.1016/j.insmatheco.2011.08.006 10.1007/BF00535637 10.1239/aap/1300198519 10.1007/s11009-017-9564-5 10.1112/plms/s3-23.1.16 10.1007/s12044-012-0085-4 10.1142/2779 10.1007/s10959-023-01285-2 |
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| Issue | 6 |
| Keywords | order statistics Extreme value theory generalized convolution Kendall type convolution Secondary 44A35 62G30 60G70 weak stability with respect to generalized convolution Williamson transform Primary 60E10 60E05 |
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| Snippet | This paper deals with the generalized convolutions connected with the Williamson transform and the maximum operation. We focus on such convolutions which can... |
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| Title | How Exceptional is the Extremal Kendall and Kendall-Type Convolution |
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