Invariance properties of the negative binomial Levy process and stochastic self-similarity

Kozubowski, Tomasz J.; Podgórski, Krzysztof

doi:10.12988/imf.2007.07133

Cited by 17 publications

(20 citation statements)

References 18 publications

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“…Solutions of similar cell division operators can be found in Michel et al (2004) and Michel (2006); da/β(a) is the mean expectation time of reaching the state a + da in a negative binomial Levy process (Kozubowski & Podgorski 2007) of parameter β(a), hence h(x) is the mean observed time between age 1 and age x.…”

Section: The Dynamical Equation Describing the Evolution Of The Distrmentioning

confidence: 93%

Synchrony in reaction–diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves

Abbas

Demongeot

Glade

2009

Phil. Trans. R. Soc. A.

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The paper presents the classical age-dependent approach of the morphogenesis in the framework of the von Foerster equation, in which we introduce a new constraint and study a new feature: (i) the new constraint concerns cell proliferation along the contour lines of the cell density, depending on the local curvature such as it favours the amplification of the concavities (like in the gastrulation process) and (ii) the new feature consists of considering, on the cell density surface, a remarkable line (the null mean Gaussian curvature line), on which the normal diffusion vanishes, favouring local coexistence of diffusing morphogens, metabolites or cells, and hence the auto-assemblages of these entities. Two applications to biological multi-agents systems are studied, gastrulation and feather morphogenesis.

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Section: The Dynamical Equation Describing the Evolution Of The Distrmentioning

confidence: 93%

Synchrony in reaction–diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves

Abbas

Demongeot

Glade

2009

Phil. Trans. R. Soc. A.

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“…This happens to be infinitely divisible distribution and thus extend to a continuous time process N p (t), t ≥ 0, see Kozubowski and Podgórski (2007) and Kozubowski and Podgórski (2009) for further details. Here just let us recall that N p (s) has a negative binomial distribution with parameters p ∈ (0, 1) and s > 0 given by…”

Section: The Negative Binomial Distributed Number Of Claimsmentioning

confidence: 99%

“…We consider here the standard gamma process for which Γ(1) has the standard exponential distribution. As shown in Kozubowski and Podgórski (2007), if a negative binomial process N p (t) is independent of Laplace motion L d the following invariance property of L d (t) holds:…”

Section: The Negative Binomial Distributed Number Of Claimsmentioning

confidence: 99%

Third cumulant for multivariate aggregate claim models

Loperfido

Mazur

Podgórski³

2017

Scandinavian Actuarial Journal

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The third moment cumulant for the aggregated multivariate claims is considered. A formula is presented for the general case when the aggregating variable is independent of the multivariate claims. It is discussed how this result can be used to obtain a formula for the third cumulant for a classical model of multivariate claims. Two important special cases are considered. In the first one, multivariate skewed normal claims are considered and aggregated by a Poisson variable. The second case is dealing with multivariate asymmetric generalized Laplace and aggregation is made by a negative binomial variable. Due to the invariance property the latter case can be derived directly leading to the identity involving the cumulant of the claims and the aggregated claims. There is a well established relation between asymmetric Laplace motion and negative binomial process that corresponds to the invariance principle of the aggregating claims for the generalized asymmetric Laplace distribution. We explore this relation and provide multivariate continuous time version of the results.

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“…Indeed, when c = 0, the function φ(k; c, d) becomes a constant independent of the current state and the A&SD model (1) is reduced to a pure age-dependent (AD) Markov chain with negative binomial independent increments. Due to the relationship between negative binomial and gamma random variables [6], this AD model can be regarded as the discrete time-discrete state version of the widely applied gamma process [2,12]. On the other hand, when b = 1, the function ψ(ν; a, b) becomes a constant independent of the current age, and the A&SD model (1) is reduced to a pure state-dependent (SD) homogeneous Markov chain, which is practically equivalent to the SD model presented in [5].…”

Section: Model Descriptionmentioning

confidence: 99%

A parametric Markov chain to model age- and state-dependent wear processes

Giorgio

Guida

Pulcini

2010

Contributions to Statistics

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Abstract. Many technological systems are subjected, during their operating life, to a gradual wear process which, in the long run, may cause failure. According to the literature, it results that statisticians and engineers have almost always modeled wear processes by independent increments models, which imply that future wear is assumed to depend, at most, on the system's age. In many cases it seems to be more realistic and appropriate to adopt stochastic models which assume that factors other than age affect wear. Indeed, wear models which can (also) account for the dependence on the system's state have been previously proposed in the literature [1,3,11,13]. Many of the abovementioned models present a very complex structure that prevents their application to the kind of data that are usually available. As such, in this paper, a new simple parametric Markov chain wear model is proposed, in which the transition probabilities between process states depend on both the current age and the current wear level of the system. An application based on a real data set referring to the wear process of the cylinder liners of heavy-duty diesel engines for marine propulsion is analysed and discussed.

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Invariance properties of the negative binomial Levy process and stochastic self-similarity

Cited by 17 publications

References 18 publications

Synchrony in reaction–diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves

Synchrony in reaction–diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves

Third cumulant for multivariate aggregate claim models

A parametric Markov chain to model age- and state-dependent wear processes

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