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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 S n 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 

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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

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