Subordinated compound Poisson processes of order k

Sengar, Ayushi S.; Upadhye, N. S.

doi:10.15559/20-vmsta165

Cited by 7 publications

(6 citation statements)

References 25 publications

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“…Further, for and , the CGCP reduces to the Poisson process. On taking for all , the CGCP reduces to the compound Poisson process of order k (see [27]). Also, for , , , the CGCP reduces to the compound Pólya–Aeppli process of order k .…”

Section: Compound Generalized Counting Processmentioning

confidence: 99%

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On a time-changed variant of the generalized counting process

Khandakar,

Kataria

2023

J. Appl. Probab.

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In this paper, we time-change the generalized counting process (GCP) by an independent inverse mixed stable subordinator to obtain a fractional version of the GCP. We call it the mixed fractional counting process (MFCP). The system of fractional differential equations that governs its state probabilities is obtained using the Z transform method. Its one-dimensional distribution, mean, variance, covariance, probability generating function, and factorial moments are obtained. It is shown that the MFCP exhibits the long-range dependence property whereas its increment process has the short-range dependence property. As an application we consider a risk process in which the claims are modelled using the MFCP. For this risk process, we obtain an asymptotic behaviour of its finite-time ruin probability when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large. Later, we discuss some distributional properties of a compound version of the GCP.

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Section: Compound Generalized Counting Processmentioning

confidence: 99%

“…The proof of Theorem 4.1 follows similar lines to that of Theorem 1 of [27], and thus it is omitted.…”

Section: Compound Generalized Counting Processmentioning

confidence: 99%

On a time-changed variant of the generalized counting process

Khandakar,

Kataria

2023

J. Appl. Probab.

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“…Recall that a doubly stochastic Poisson process is a stochastic point process of the form N(t) def = Π(L(t)), where Π(t), t ≥ 0 is a Poisson process with unit intensity and the stochastic process L(t), t ≥ 0 is independent of Π(t) and possesses the following properties: L(0) = 0, P(L(t) < ∞) = 1 for any t > 0, the sample paths of L(t) are right-continuous and do not decrease. For more details concerning Cox and more general subordinated processes see, e.g., [34][35][36].…”

Section: Generalized Burr Distribution As a Limit Law For Extreme Ord...mentioning

confidence: 99%

Mixture Representations for Generalized Burr, Snedecor–Fisher and Generalized Student Distributions with Related Results

Korolev,

Zeifman

2023

Mathematics

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In this paper, the representability of the generalized Student’s distribution as uniform and normal-scale mixtures is considered. It is also shown that the generalized Burr and the Snedecor–Fisher distributions can be represented as the scale mixtures of uniform, folded normal, exponential, Weibull or Fréchet distributions. New multiplication-type theorems are proven for these and related distributions. The relation between the generalized Student and generalized Burr distribution is studied. It is shown that the Snedecor–Fisher distribution is a special case of the generalized Burr distribution. Based on these mixture representations, some limit theorems are proven for random sums in which the symmetric and asymmetric generalized Student or symmetric and asymmetric two-sided generalized Burr distributions are limit laws. Also, limit theorems are proven for maximum and minimum random sums and absolute values of random sums in which the generalized Burr distributions are limit laws.

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“…Point processes involving integer valued random variables have been widely used in modeling of high frequency financial data see [1][2][3]. In 1946, J.G.…”

Section: Introductionmentioning

confidence: 99%

Time-changed Fractional Convoluted Skellam Process

Sengar

Upadhye

2022

Preprint

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In this article, we introduce a convoluted version of Skellam process. We also give an alternate definition through its transition probabilities and discuss its distributional properties. Next, we consider the fractional version of convoluted Skellam process, which we call as fractional convoluted Skellam process (FCSP), and derive the probability generating function, mean, variance, covariance function, and establish its the long-range dependence property. Later, we introduce two time-changed variants of fractional convoluted Skellam process, in which we introduce time-change FCSP by Lévy subordinator and its first exit time, and call as TCFCSP-I and TCFCSP-II respectively. We also derive the law of iterated logarithm property for TCFCSP-I. Later we give the expression for the asymptotic behavior of moments of TCFCSP-II. MSC Classification: 60G22; 60G55

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Subordinated compound Poisson processes of order k

Cited by 7 publications

References 25 publications

On a time-changed variant of the generalized counting process

On a time-changed variant of the generalized counting process

Mixture Representations for Generalized Burr, Snedecor–Fisher and Generalized Student Distributions with Related Results

Time-changed Fractional Convoluted Skellam Process

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