Counting processes with Bernštein intertimes and random jumps
Enzo Orsingher,Bruno Toaldo +1 more
TLDR
In this paper, the authors considered point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν.Abstract:
In this paper we consider point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν. We obtain the general expression of the probability generating functions Gf of Nf, the equations governing the state probabilities pkf of Nf, and their corresponding explicit forms. We also give the distribution of the first-passage times Tkf of Nf, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times τjlj of jumps with height lj (∑j=1rlj = k) under the condition N(t) = k for all these special processes is investigated in detail.read more
Citations
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A note on Hadamard fractional differential equations with varying coefficients and their applications in probability
TL;DR: In this paper, the authors show connections between special functions arising from generalized COM-Poisson type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators.
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Time-changed Poisson processes of order k
TL;DR: In this article, the Poisson process of order k (PPoK) time-changed with an independent Levy subordinator and its inverse was studied, which they called TCPPoK-I and TCPPoK-II.
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Some probabilistic properties of fractional point processes
TL;DR: In this paper, the first hitting times of generalized Poisson processes Nf(t) related to Bernstein functions f are studied and the hitting probabilities P{Tαk < ∞} are explicitly obtained and analyzed.
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Compound Poisson process with a Poisson subordinator
TL;DR: In this paper, the first-crossing-time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where they provide some closed-form results, and (ii) a linearly increasing boundary.
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On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators
Khrystyna Buchak,Lyudmyla Sakhno +1 more
TL;DR: In this paper, the governing equations for marginal distributions of Poisson and Skellam processes were presented in terms of convolution-type derivatives, and the equations were given by using inverse subordinators.
References
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Book
Bernstein Functions: Theory and Applications
TL;DR: In this paper, the authors present a self-contained and unified approach to Bernstein functions and closely related function classes, bringing together old and establishing new connections, and an extensive list of complete Bernstein functions with their representations is provided.
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Fractional Poisson process
TL;DR: In this article, a fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov-Feller equation.
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The Fractional Poisson Process and the Inverse Stable Subordinator
TL;DR: In this article, it was shown that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional poisson process with Mittag-Leffler waiting times, which unifies the two main approaches in stochastic theory of time-fractional diffusion equations.
Book
The Stability of Matter: From Atoms to Stars
TL;DR: In this paper, the peculiar mechanics of the elementary particles (electrons and nuclei) that constitute ordinary matter so that the material world can have both rich variety and stability are discussed.
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Generalized gamma measures and shot-noise Cox processes
TL;DR: In this paper, a parametric family of completely random measures, which includes gamma random measures and positive stable random measures as well as inverse Gaussian measures, is defined and used in a shot-noise construction as intensity measures for Cox processes.