On Distributions of Certain State-Dependent Fractional Point Processes

On Distributions of Certain State-Dependent Fractional Point Processes J Theor Probab https://doi.org/10.1007/s10959-018-0835-z On Distributions of Certain State-Dependent Fractional Point Processes 1 1 K. K. Kataria · P. Vellaisamy Received: 16 January 2018 / Revised: 13 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We obtain the explicit expressions for the state probabilities of various state-dependent versions of fractional point processes. The inversion of the Laplace transforms of the state probabilities of such processes is rather cumbersome and involved. We employ the Adomian decomposition method to solve the difference- differential equations governing the state probabilities of these state-dependent processes. The distributions of some convolutions of the Mittag-Leffler random vari- ables are derived as special cases of the obtained results. Keywords State-dependent fractional pure birth process · State-dependent time fractional Poisson process · Time fractional Poisson process Mathematics Subject Classification (2010) 60G22 · 60G55 1 Introduction The Poisson process is an important counting process which has applications in several fields. A characterization of the Poisson process can be given in terms of Kolmogorov equations. A stochastic process {N (t,λ)} is said to be a Poisson process with t ≥0 intensity λ> 0 if the process has independent and stationary increments, and the state B http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Theoretical Probability Springer Journals

On Distributions of Certain State-Dependent Fractional Point Processes

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Probability Theory and Stochastic Processes; Statistics, general
ISSN
0894-9840
eISSN
1572-9230
D.O.I.
10.1007/s10959-018-0835-z
Publisher site
See Article on Publisher Site

Abstract

J Theor Probab https://doi.org/10.1007/s10959-018-0835-z On Distributions of Certain State-Dependent Fractional Point Processes 1 1 K. K. Kataria · P. Vellaisamy Received: 16 January 2018 / Revised: 13 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We obtain the explicit expressions for the state probabilities of various state-dependent versions of fractional point processes. The inversion of the Laplace transforms of the state probabilities of such processes is rather cumbersome and involved. We employ the Adomian decomposition method to solve the difference- differential equations governing the state probabilities of these state-dependent processes. The distributions of some convolutions of the Mittag-Leffler random vari- ables are derived as special cases of the obtained results. Keywords State-dependent fractional pure birth process · State-dependent time fractional Poisson process · Time fractional Poisson process Mathematics Subject Classification (2010) 60G22 · 60G55 1 Introduction The Poisson process is an important counting process which has applications in several fields. A characterization of the Poisson process can be given in terms of Kolmogorov equations. A stochastic process {N (t,λ)} is said to be a Poisson process with t ≥0 intensity λ> 0 if the process has independent and stationary increments, and the state B

Journal

Journal of Theoretical ProbabilitySpringer Journals

Published: Jun 5, 2018

References

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