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
More filters
Journal ArticleDOI
Time-changed space-time fractional Poisson process
K. K. Kataria,M. Khandakar +1 more
TL;DR: In this paper, a time-changed version of the space-time fractional Poisson process (STFPP) by time changing it by an independent Levy subordinator with finite moments of any
Posted Content
Fractional non-homogeneous Poisson and P\'olya-Aeppli processes of order $k$ and beyond
TL;DR: In this article, a fractional non-homogeneous Poisson Poisson process of order $k$ was introduced and two nonhomogeneous polya-aeppli processes were characterized by deriving their non-local governing equations.
Posted Content
Superposition of time-changed Poisson processes and their hitting times
TL;DR: In this paper, the authors studied some extensions of the Poisson process of order $i$ for different forms of weights and also with the time-changed versions, with Bern\v stein subordinator playing the role of time.
Journal ArticleDOI
A simple proof of the Lévy–Khintchine formula for subordinators
TL;DR: In this paper, the authors present a relatively simple and mostly elementary proof of the Levy-Khintchine formula for subordinators, which is a compound Poisson process which is easy to investigate using elementary probabilistic techniques, such as conditional expectations, probability generating function and convergence of discrete random variables.
Posted Content
Fractional Skellam Process of Order $k$
K. K. Kataria,M. Khandakar +1 more
TL;DR: In this article, a fractional version of the Skellam process of order $k$ by time-changing it with an independent inverse stable subordinator is introduced and an integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained.
References
More filters
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.
Journal ArticleDOI
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.
Journal ArticleDOI
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.
Journal ArticleDOI
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.