Lithuanian Mathematical Journal

, Volume 55, Issue 4, pp 518–542 | Cite as

Classical definitions of the Poisson process do not coincide in the case of generalized convolutions

  • Barbara H. Jasiulis-Gołdyn
  • Jolanta K. Misiewicz
Article

Abstract

In the paper, we consider a generalization of the notion of Poisson process to the case where the classical convolution is replaced by the generalized convolution in the sense of Urbanik [K. Urbanik, Generalized convolutions, Stud. Math., 23:217–245, 1963] following two classical definitions of the Poisson process. First, for every generalized convolution ◊, we define a ◊-generalized Poisson process of type I as a Markov process with ◊-generalized Poisson distribution. Such processes have stationary independent increments in the sense of generalized convolution, but usually they do not live on ℕ0. A ◊-generalized Poisson process of type II is defined as a renewal process based on the sequence Sn that is a Markov process with step distribution with the lack-of-memory property. Such processes take values in ℕ0; however, they must not be Markov processes or have independent increments, even in the generalized convolution sense. It turns out that the second construction is possible only for monotonic generalized convolutions that admit the existence of distributions with lack of memory, and thus we also study these properties.

Keywords

generalized convolution generalized random walk lack-of-memory property Markov process Poisson process renewal process 

MSC

primary 60E07 secondary 44A35 60K05 60J05 60E10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N.H. Bingham, On a theorem of Kłosowska about generalised convolutions, Colloq. Math., 48(1):117–125, 1984.MATHMathSciNetGoogle Scholar
  2. 2.
    M. Borowiecka-Olszewska, B.H. Jasiulis-Gołdyn, J.K. Misiewicz, and J. Rosiński, Lévy processes and stochastic integral in the sense of generalized convolution, Bernoulli, 21(4):2513–2551, 2015.MathSciNetCrossRefGoogle Scholar
  3. 3.
    B.H. Jasiulis, Limit property for regular and weak generalized convolutions, J. Theor. Probab., 23(1):315–327, 2010.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    B.H. Jasiulis-Gołdyn, Kendall random walks, Probab. Math. Stat., 2015 (to appear).Google Scholar
  5. 5.
    B.H. Jasiulis-Gołdyn and A. Kula, The Urbanik generalized convolutions in the non-commutative probability and a forgotten method of constructing generalized convolution, Proc. Indian Acad. Sci., Math. Sci., 122(3):437–458, 2012.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    B.H. Jasiulis-Gołdyn and J.K. Misiewicz, On the uniqueness of the Kendall generalized convolution, J. Theor. Probab., 24(3):746–755, 2011.MATHCrossRefGoogle Scholar
  7. 7.
    B.H. Jasiulis-Gołdyn and J.K.Misiewicz, Weak Lévy–Khintchne representation for weak infinite divisibility, Theory Probab. Appl., 2015 (to appear).Google Scholar
  8. 8.
    F. Killmann and E. von Collani, A note on the convolution of the uniform and related distributions and their use in quality control, Econ. Qual. Control, 16(1):17–41, 2001.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    J.F.C. Kingman, Random walks with spherical symmetry, Acta Math., 109(1):11–53, 1963.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    J. Kucharczak and K. Urbanik, Transformations preserving weak stability, Bull. Pol. Acad. Sci., Math., 34(7–8): 475–486, 1986.MATHMathSciNetGoogle Scholar
  11. 11.
    A. McNeil and J. Neslehová, Multivariate Archimedean copulas, d-monotone functions and ℓ1-norm symmetric distributions, Ann. Stat., 37(5B):3059–3097, 2009.MATHCrossRefGoogle Scholar
  12. 12.
    J.K. Misiewicz, Weak stability and generalized weak convolution for random vectors and stochastic processes, in D. Denteneer, F. den Hollander, and E. Verbitskiy (Eds.), Dynamics and Stochastics. Festschrift in honor of M.S. Keane, IMS Lect. Notes, Monogr. Ser., Vol. 48, Institute of Mathematical Statistics, Beachwood, OH, 2006, pp. 109–118.Google Scholar
  13. 13.
    J.K. Misiewicz, K. Oleszkiewicz, and K. Urbanik, Classes of measures closed under mixing and convolution.Weak stability, Stud. Math., 167(3):195–213, 2005.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    N.V. Thu, Generalized independent increments processes, Nagoya Math. J., 133:155–175, 1994.MATHMathSciNetGoogle Scholar
  15. 15.
    N.V. Thu, A Kingman convolution approach to Bessel process, Probab. Math. Stat., 29(1):119–134, 2009.MATHGoogle Scholar
  16. 16.
    K. Urbanik, Generalized convolutions, Stud. Math., 23:217–245, 1963.MathSciNetGoogle Scholar
  17. 17.
    K. Urbanik, Generalized convolutions. II, Stud. Math., 45:57–70, 1973.MATHMathSciNetGoogle Scholar
  18. 18.
    K. Urbanik, Remarks on B-stable probability distributions, Bull. Acad. Pol. Sci., Math., 24(9):783–787, 1976.MATHMathSciNetGoogle Scholar
  19. 19.
    K. Urbanik, Generalized convolutions. III, Stud. Math., 80(2):167–189, 1984.MATHMathSciNetGoogle Scholar
  20. 20.
    K. Urbanik, Generalized convolutions. IV, Stud. Math., 83(1):57–95, 1986.MATHMathSciNetGoogle Scholar
  21. 21.
    K. Urbanik, Quasi-regular generalized convolutions, Colloq. Math., 55(1):147–162, 1988.MATHMathSciNetGoogle Scholar
  22. 22.
    C. Vignat and A. Plastimo, Geometry of the central limit theorem in the nonextensive case, Phys. Lett., A, 373(20):1713–1718, 2009.MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    V. Vol’kovich, Multidimensional B-stable distributions and some generalized convolutions, in V.M. Zolotarev and V.V. Kalashnikov (Eds.), Stability Problems of Stochastic Models. Proceedings of VNIISI Seminar, Vol. M, VNIISI, Moscow, 1984, pp. 40–53 (in Russian).Google Scholar
  24. 24.
    V. Vol’kovich, On symmetric stochastic convolutions, J. Theor. Probab., 5(3):417–430, 1992.MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    V. Vol’kovich, D. Toledano-Ketai, and R. Avros, On analytical properties of generalized convolutions, Banach Center Publications. Stability in Probability, 5(3):243–274, 2010.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Barbara H. Jasiulis-Gołdyn
    • 1
  • Jolanta K. Misiewicz
    • 2
  1. 1.Institute of MathematicsUniversity of WrocławWrocławPoland
  2. 2.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

Personalised recommendations