Page 116 - Special Topic Session (STS) - Volume 3

STS518 B.H. Jasiulis G. et al.

Infinitely divisible probability measures under
generalized convolutions
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B.H. Jasiulis - Gołdyn ; M. Arendarczyk ; M. Borowiecka-Olszewska ; J.K.
Misiewicz ; E. Omey ; J. Rosiński
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1 Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384
Wrocław, Poland
Faculty of Mathematics, Computer Science and Econometrics, University of
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Zielona Góra, ul. Prof. Z. Szafrana 4A, 65-516 Zielona Góra, Poland
3 Faculty of Mathematics and Information Science, Warsaw University of
Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland
KU Leuven, Warmoesberg 26, 1000 Brussels, Belgium
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5 Department of Mathematics, 227 Ayres Hall, University of Tennessee,
Knoxville TN 37996, USA

Abstract
Kingman, in his seminal work [13], introduced a new type of convolution of
distributions that is naturally related to spherically symmetric random walks.
Motivated by this paper, Urbanik in a series of papers [17] established a theory
of generalized convolutions ⋄ as certain binary commutative and associative
operations that include classical and Kingman’s convolutions as a special case.
This theory was further developed by Bingham ([2, 3]) in the context of
regularly varying functions. There is a rich class of examples of generalized
convolutions that are motivated by problems in applications of probability
theory. For instance, the distribution of the maximum of two independent
random variables is a generalized convolution fundamentally associated with
the extreme value theory, and extensively applied to model events that rarely
occur, but the appearance of which causes large losses. Similarly, to the
classical theory, we define infinite divisibility with respect to generalized
convolution ⋄ and establish Lévy-Khintchine representation [11]. Lévy and
additive stochastic processes under generalized convolutions are constructed
as the Markov processes in ([5]). In this paper we survey examples of
generalized convolutions and related Lévy-Khintchine representation. Results
on Kendall convolution and extreme Markov chains driven by the Kendall
convolution ([1, 5, 10]) using Williamson transform ([18]) are also presented.

Keywords
infinitely divisible probability measure; generalized convolution; Kendall
random walk; LévyKchintchine representation; regular variation

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