On analytical properties of generalized convolutions

Volkovich, Zeev; Toledano-Kitai, Dvora; Avros, Renata

doi:10.4064/bc90-0-16

Cited by 5 publications

(4 citation statements)

References 20 publications

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“…Notice that the operation △ α is a generalized convolution in the sense introduced by Urbanik (see [12,29,30]) having the following properties:…”

Section: Williamson Transform and Kendall Convolutionmentioning

confidence: 99%

Fluctuations of extremal Markov chains driven by the Kendall convolution

Jasiulis-Gołdyn¹,

Omey²,

Staniak³

2019

Preprint

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The paper deals with fluctuations of Kendall random walks, which are extremal Markov chains. We give the joint distribution of the first ascending ladder epoch and height over any level a ≥ 0 and distribution of maximum and minimum for these extremal Markovian sequences. We show that distribution of the first crossing time of level a ≥ 0 is a mixture of geometric and negative binomial distributions. The Williamson transform is the main tool for considered problems connected with the Kendall convolution.

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“…Notice that the operation △ α is a generalized convolution in the sense introduced by Urbanik (see [12,29,30]) having the following properties:…”

Section: Williamson Transform and Kendall Convolutionmentioning

confidence: 99%

Fluctuations of extremal Markov chains driven by the Kendall convolution

Jasiulis-Gołdyn¹,

Omey²,

Staniak³

2019

Preprint

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“…Notice that properties of Kendall convolution together with the one described in the last proposition show that the Kendall convolution is an example of generalized convolutions in the sense defined by K. Urbanik. More about generalized convolutions one can find in [2,5,6,7,8,9,10,11].…”

Section: Definition 22 By the Kendall Convolution We Understand Opementioning

confidence: 99%

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

Jasiulis-Gołdyn

Misiewicz

2017

Lith Math J

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“…One can find many open problems connected with generalized convolutions (e.g. in [26]). All the basic information about the examples listed bellow one can find in [1,7,8,10,20,19,18,17,16,24] Examples.…”

Section: Urbanik's Generalized Convolutionsmentioning

confidence: 99%

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

Jasiulis-Gołdyn

Misiewicz

2015

Lith Math J

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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.

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On analytical properties of generalized convolutions

Cited by 5 publications

References 20 publications

Fluctuations of extremal Markov chains driven by the Kendall convolution

Fluctuations of extremal Markov chains driven by the Kendall convolution

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

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

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