Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-17T08:22:16.718Z Has data issue: false hasContentIssue false
  • English
  • Français

Correlated fractional counting processes on a finite-time interval

Part of: Other special functions Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  30 March 2016

Luisa Beghin ,
Roberto Garra  and
Claudio Macci
Luisa Beghin*
Affiliation:
Sapienza University of Rome
Roberto Garra*
Affiliation:
Sapienza University of Rome
Claudio Macci*
Affiliation:
University of Rome Tor Vergata
*
∗Postal address: Dipartimento di Scienze Statistiche, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy. Email address: [email protected]
∗∗Postal address: Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 16, I-00161 Roma, Italy. Email address: [email protected]
∗∗∗Postal address: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present some correlated fractional counting processes on a finite-time interval. This will be done by considering a slight generalization of the processes in Borges et al. (2012). The main case concerns a class of space-time fractional Poisson processes and, when the correlation parameter is equal to 0, the univariate distributions coincide with those of the space-time fractional Poisson process in Orsingher and Polito (2012). On the one hand, when we consider the time fractional Poisson process, the multivariate finite-dimensional distributions are different from those presented for the renewal process in Politi et al. (2011). We also consider a case concerning a class of fractional negative binomial processes.

Keywords

Poisson processnegative binomial processweighted process

MSC classification

Primary: 33E12: Mittag-Leffler functions and generalizations 60G22: Fractional processes, including fractional Brownian motion 60G55: Point processes
Secondary: 60E05: Distributions
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1045 - 1061
Copyright
Copyright © Applied Probability Trust 2015 

References

[1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau Standards Appl. Math. Ser. 55). U.S. Government Printing Office, Washington, D. C. Google Scholar
[2] Angulo, J. M., Ruiz-Medina, M. D., Anh, V. V. and Grecksch, W. (2000). Fractional diffusion and fractional heat equation. Adv. Appl. Prob. 32, 1077–1099.Google Scholar
[3] Baeumer, B., Meerschaert, M. M. and Nane, E. (2009). Brownian subordinators and fractional Cauchy problems. Trans. Amer. Math. Soc. 361, 3915–3930.Google Scholar
[4] Balakrishnan, N. and Kozubowski, T. J. (2008). A class of weighted Poisson processes. Statist. Prob. Lett. 78, 2346–2352.CrossRefGoogle Scholar
[5] Beghin, L. and D'Ovidio, M. (2014). Fractional Poisson process with random drift. Electron. J. Prob. 19, 26 pp.Google Scholar
[6] Beghin, L. and Macci, C. (2014). Fractional discrete processes: compound and mixed Poisson representations. J. Appl. Prob. 51, 19–36.Google Scholar
[7] Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electron. J. Prob. 14, 1790–1827.Google Scholar
[8] Beghin, L. and Orsingher, E. (2010). Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Prob. 15, 684–709.Google Scholar
[9] Borges, P., Rodrigues, J. and Balakrishnan, N. (2012). A class of correlated weighted Poisson processes. J. Statist. Planning Inference 142, 366–375.Google Scholar
[10] Jarad, F., Abdeljawad, T. and Baleanu, D. (2012). Caputo-type modification of the Hadamard fractional derivatives. Adv. Difference Equations 2012, 8 pp.Google Scholar
[11] Johnson, N. L., Kotz, S. and Kemp, A. W. (1992). Univariate Discrete Distributions , 2nd edn. John Wiley, New York.Google Scholar
[12] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.Google Scholar
[13] Kostinski, A. B. (2001). On the extinction of radiation by a homogeneous but spatially correlated random medium. J. Optical Soc. Amer. A 18, 1929–1933.Google Scholar
[14] Kozubowski, T. J. and PodgóRski, K. (2009). Distributional properties of the negative binomial Lévy process. Prob. Math. Statist. 29, 43–71.Google Scholar
[15] Kumar, A., Nane, E. and Vellaisamy, P. (2011). Time-changed Poisson processes. Statist. Prob. Lett. 81, 1899–1910.Google Scholar
[16] Laskin, N. (2003). Fractional Poisson process. Commun. Nonlinear Sci. Numer. Simul. 8, 201–213.CrossRefGoogle Scholar
[17] Lefevre, C. and Picard, P. (2011). A new look at the homogeneous risk model. Insurance Math. Econom. 49, 512–519.CrossRefGoogle Scholar
[18] Mainardi, F. (1996). The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9, 23–28.Google Scholar
[19] Mainardi, F., Gorenflo, R. and Scalas, E. (2004). A fractional generalization of the Poisson processes. Vietnam J. Math. 32, 53–64.Google Scholar
[20] Meerschaert, M. M., Nane, E. and Vellaisamy, P. (2011). The fractional Poisson process and the inverse stable subordinator. Electron. J. Prob. 16, 1600–1620.CrossRefGoogle Scholar
[21] Orsingher, E. and Beghin, L. (2004). Time-fractional telegraph equations and telegraph processes with Brownian time. Prob. Theory Relat. Fields 128, 141–160.CrossRefGoogle Scholar
[22] Orsingher, E. and Beghin, L. (2009). Fractional diffusion equations and processes with randomly-varying time. Ann. Prob. 37, 206–249.CrossRefGoogle Scholar
[23] Orsingher, E. and Polito, F. (2010). Fractional pure birth processes. Bernoulli 16, 858–881.Google Scholar
[24] Orsingher, E. and Polito, F. (2012). The space-fractional Poisson process. Statist. Prob. Lett. 82, 852–858.Google Scholar
[25] Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego, CA.Google Scholar
[26] Politi, M., Kaizoji, T. and Scalas, E. (2011). Full characterization of the fractional Poisson process. Europhys. Lett. 96, 20004.Google Scholar
[27] Schneider, W. R. and Wyss, W. (1989). Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144.Google Scholar
[28] Vellaisamy, P. and Maheshwari, A. (2014). Fractional negative binomial and Polya processes. Preprint. Available at http://arxiv.org/abs/1306.2493.Google Scholar