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Aequationes mathematicae

, Volume 92, Issue 5, pp 911–933 | Cite as

Generalized convolutions and the Levi-Civita functional equation

  • J. K. Misiewicz
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Abstract

In Borowiecka et al. (Bernoulli 21(4):2513–2551, 2015) the authors show that every generalized convolution can be used to define a Markov process, which can be treated as a Lévy process in the sense of this convolution. The Bessel process is the best known example here. In this paper we present new classes of regular generalized convolutions enlarging the class of such Markov processes. We give here a full characterization of such generalized convolutions \(\diamond \) for which \(\delta _x \diamond \delta _1\), \(x \in [0,1]\), is a convex linear combination of \(n=3\) fixed measures and only the coefficients of the linear combination depend on x. For \(n=2\) it was shown in Jasiulis-Goldyn and Misiewicz (J Theor Probab 24(3):746–755, 2011) that such a convolution is unique (up to the scale and power parameters). We show also that characterizing such convolutions for \(n \geqslant 3\) is equivalent to solving the Levi-Civita functional equation in the class of continuous generalized characteristic functions.

Keywords

Generalized convolution Kendall convolution Levi-Civita functional equation 

Mathematics Subject Classification

Primary 60E05 39B22 Secondary 60E10 

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

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