Classical definitions of the Poisson process do not coincide in the case of generalized convolutions
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.
Keywordsgeneralized convolution generalized random walk lack-of-memory property Markov process Poisson process renewal process
MSCprimary 60E07 secondary 44A35 60K05 60J05 60E10
Unable to display preview. Download preview PDF.
- 4.B.H. Jasiulis-Gołdyn, Kendall random walks, Probab. Math. Stat., 2015 (to appear).Google Scholar
- 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
- 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
- 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