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Counting processes with Bernštein intertimes and random jumps

Published online by Cambridge University Press:  30 March 2016

Enzo Orsingher*
Affiliation:
Sapienza Università di Roma
Bruno Toaldo*
Affiliation:
Sapienza Università di Roma
*
Postal address: Dipartimento di Statistica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy.
Postal address: Dipartimento di Statistica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy.
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Abstract

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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 Bernštein functions f with Lévy measure v. 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 of jumps with height lj () under the condition N(t) = k for all these special processes is investigated in detail.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2015 

References

Beghin, L. (2015). Fractional gamma and gamma-subordinated processes. Stoch. Anal. Appl. 33, 903926.CrossRefGoogle Scholar
Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. Appl. Prob. 31, 929953.CrossRefGoogle Scholar
Cahoy, D. O. and Polito, F. (2012). On a fractional binomial process. J. Statist. Phys. 146, 646662.CrossRefGoogle Scholar
Di Crescenzo, A., Martinucci, B. and Zacks, S. (2015). Compound Poisson process with a Poisson subordinator. J. Appl. Prob. 52, 360374.CrossRefGoogle Scholar
Kozubowski, T. J. and PodgóRski, K. (2009). Distributional properties of the negative binomial Lévy process. Prob. Math. Statist. 29, 4371.Google Scholar
Kreer, M., Kizilersü, A. and Thomas, A. W. (2014). Fractional Poisson processes and their representation by infinite systems of ordinary differential equations. Statist. Prob. Lett. 84, 2732.CrossRefGoogle Scholar
Kumar, A., Nane, E. and Vellaisamy, P. (2011). Time-changed Poisson processes. Statist. Prob. Lett. 81, 18991910.CrossRefGoogle Scholar
Laskin, N. (2003). Fractional Poisson process. Chaotic transport and complexity in classical and quantum dynamics. Commun. Nonlinear Sci. Numer. Simul. 8, 201213.CrossRefGoogle Scholar
Lieb, E. H. (1990). The stability of matter: from atoms to stars. Bull. Amer. Math. Soc. 22, 149.CrossRefGoogle Scholar
Meerschaert, M. M., Nane, E. and Vellaisamy, P. (2011). The fractional Poisson process and the inverse stable subordinator. Electron. J. Prob. 16, 16001620.CrossRefGoogle Scholar
Orsingher, E. and Polito, F. (2012a). Compositions, random sums and continued random fractions of Poisson and fractional Poisson processes. J. Statist. Phys. 148, 233249.CrossRefGoogle Scholar
Orsingher, E. and Polito, F. (2012b). The space-fractional Poisson process. Statist. Prob. Lett. 82, 852858.CrossRefGoogle Scholar
Schilling, R. L., Song, R. and Vondracek, Z. (2010). Bernstein Functions: Theory and Applications. De Gruyter, Berlin.Google Scholar
Vellaisamy, P. and Maheshwari, A. (2014). Fractional negative binomial and Polya processes. Preprint. Available at http://arXiv.org/abs/1306.2493.Google Scholar
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