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

Abstract

Abstract. In the paper we consider a generalizations of the notion of Poisson process to the case when classical convolution is replaced by generalized convolution in the sense of K. Urbanik [16] following two classical definitions of Poisson process. First, for every generalized convolution ⋄ we define ⋄-generalized Poisson process type I as a Markov process with the ⋄-generalized Poisson distribution. Such processes have stationary independent increments in the sense of generalized convolution, but usually they do not live on N 0 . The ⋄-generalized Poisson process type II is defined as a renewal process based on the sequence S n , which is a Markov process with the step with the lack of memory property. Such processes take values in N 0 , however they do not have to be Markov processes, do not have to have independent increments, even in generalized convolution sense. It turns out that the second construction is possible only for monotonic generalized convolutions which admit the existence of distributions with lack of memory, thus we also study these properties.

Cramer-Lundberg model for some classes of extremal Markov sequences

“…We assume here that the variables V k , responsible for the insurance premium during the time T k are independent identically distributed with the cumulative distribution function F V (x) = 1−e −γx α , for x > 0. This assumption seems to be natural, since this is the distribution with the lack of memory property (see [13]) for * α -convolution. Consequently…”

“…Usually we take the random variables V k with the distribution having the lack of memory property, which in the case of ∞-convolution is given by the cumulative distribution function G(x) = 1 (a,∞) (x) for some a > 0 (see [13] for details). In this case we have…”

“…In our case we have E(u ⊕ Y t ) α − EX α t > 0 for all t > 0. First we calculate EX α t assuming that the distribution of U 1 is absolutely continuous with respect to the Lebesgue measure (if this is not the case we shall add the atomic part): In the similar way for absolutely continuous distribution of V 1 we obtain If we consider as ν the distribution with the lack of memory property for the Kendall convolution (see [13]) then for some c > 0 we have…”

Asymptotic properties of extremal Markov processes driven by Kendall convolution

“…The basic properties of the transition probability kernel are also proved. We refer to [14,17,19,18] for discussions and proofs of further properties of the process {X n }. The structure of the processes considered here (see Definition 2.6) is similar to the first order autoregressive maximal Pareto processes [2,3,24,31], max-AR(1) sequences [1], minification processes [23,24], the max-autoregressive moving average processes MARMA [9], pARMAX and pRARMAX processes [10].…”

Fluctuations of extremal Markov chains driven by the Kendall convolution

“…where measure ν ∈ P s is called the step distribution. Construction and some basic properties of this particular process are described in [4,11,13,14].…”