Lévy processes and stochastic integrals in the sense of generalized convolutions

2015

Lith Math J

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.

Section: ⋄-Generalized Poisson Process Of Type Imentioning

confidence: 69%

mentioning

confidence: 99%

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

2015

Lith Math J

“…Slightly different direct construction was given in [3]. We recall here one more definition of the Kendall random walk following [1], where only multidimensional distributions of this process are considered. This approach is more convenient in studying positive and negative excursions, which we consider in this section.…”

Section: Kendall Random Walkmentioning

confidence: 99%

“…In this paper we study positive and negative excursions for the Kendall random walk {X n : n ∈ N 0 } which can be defined by the following recurrence construction: The Kendall random walk is an example of discrete time Lévy random walk under generalized convolution studied in [1]. Some basic properties of this particular process are described in [3].…”

Section: Introductionmentioning

confidence: 99%

Kendall random walk,Williamson transform, and the corresponding Wiener–Hopf factorization

2017

Lith Math J

“…It is the main reason why it produces heavy tailed distributions. Theory of stochastic processes under generalized convolutions was introduced in [8] and following this paper we develop here renewal theory for a class of Markov chains generated by generalized convolutions. In many results we focus on the Kendall convolution case.…”

Section: Introductionmentioning

confidence: 99%

Renewal theory for extremal Markov sequences of Kendall type

Stochastic Processes and their Applications

Naskręt

et al. 2020

The paper deals with renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes associated with generalized convolutions. We prove an analogue of the Fredholm theorem for all regular generalized convolutions algebras. Using regularly varying functions we prove a Blackwell theorem for renewal processes defined by Kendall random walks.