Aequationes mathematicae

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

Generalized convolutions and the Levi-Civita functional equation

  • J. K. MisiewiczEmail author
Open Access


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.


Generalized convolution Kendall convolution Levi-Civita functional equation 

Mathematics Subject Classification

Primary 60E05 39B22 Secondary 60E10 


  1. 1.
    Borowiecka Olszewska, M., Jasiulis-Goldyn, B., Misiewicz, J.K., Rosinski, J.: Levy processes and stochastic integrals in the sense of generalized convolutions. Bernoulli 21(4), 2513–2551 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Jasiulis-Gołdyn, B.H.: On the random walk generated by the Kendall convolution. Probab. Math. Stat. 36(1), 165–185 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Jasiulis-Gołdyn, B.H., Misiewicz, J.K.: On the uniqueness of the Kendall generalized convolution. J. Theor. Probab. 24(3), 746–755 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Jasiulis-Gołdyn, B.H., Misiewicz, J.K.: Weak Lévy-Khintchine representation for weak infinite divisibility. Theory Probab. Appl. 60(1), 131–149 (2015)zbMATHGoogle Scholar
  5. 5.
    Kingman, J.F.C.: Random walks with spherical symmetry. Acta Math. 109(1), 11–53 (1963)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kucharczak, J., Urbanik, K.: Transformations preserving weak stability. Bull. Polish Acad. Sci. Math. 34(7–8), 475–486 (1986)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Misiewicz, J.K.: Weak stability and generalized weak convolution for random vectors and stochastic processes. IMS Lecture Notes-Monoghaph Series Dynamics & Stochastics 48, 109–118 (2006)Google Scholar
  8. 8.
    Székelyhidi: Convolution Type Functional Equations on Topological Abelian Groups. World Scientific, Singapore (1991). ISBN 981-02-0658-5Google Scholar
  9. 9.
    Van Thu, N.: Generalized independent increments processes. Nagoya Math. J. 133, 155–175 (1994)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Van Thu, N.: A Kingman convolution approach to Bessel Process. Probab. Math. Statist. 29(1), 119–134 (2009)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Urbanik, K.: Generalized convolutions. Studia Math. 23, 217–245 (1964)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Urbanik, K.: Generalized convolutions II. Studia Math. 45, 57–70 (1973)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Urbanik, K.: Remarks on \(\cal{B}\)-stable probability distributions. Bull. Pol. Acad. Sci. Math. 24(9), 783–787 (1976)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Urbanik, K.: Generalized convolutions III. Studia Math. 80, 167–189 (1984)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Urbanik, K.: Generalized convolutions IV. Studia Math. 83, 57–95 (1986)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Urbanik, K.: Anti-irreducible probability measures. Probab. Math. Statist. 14(1), 89–113 (1993)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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

Personalised recommendations