Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T21:24:25.756Z Has data issue: false hasContentIssue false

Fractional Discrete Processes: Compound and Mixed Poisson Representations

Published online by Cambridge University Press:  30 January 2018

Luisa Beghin*
Affiliation:
Sapienza Università di Roma
Claudio Macci*
Affiliation:
Università di Roma 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 Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Roma, 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 consider two fractional versions of a family of nonnegative integer-valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Pólya-Aeppli process, the Poisson inverse Gaussian process, and the negative binomial process. We also define and study some more general fractional versions with two fractional parameters.

Type
Research Article
Copyright
© Applied Probability Trust 

References

Allouba, H. and Zheng, W. (2001). Brownian-time processes: the PDE connection and the half-derivative generator. Ann. Prob. 29, 17801795.Google Scholar
Beghin, L. (2012). Fractional relaxation equations and Brownian crossing probabilities of a random boundary. Adv. Appl. Prob. 44, 479505.CrossRefGoogle Scholar
Beghin, L. and Macci, C. (2012). Alternative forms of compound fractional Poisson processes. Abstr. Appl. Anal. 2012, Article ID 747503, 30 pp.Google Scholar
Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar motions. Electron. J. Prob. 14, 17901827.Google Scholar
Beghin, L. and Orsingher, E. (2010). Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Prob. 15, 684709.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Heyde, C. C. and Leonenko, N. N. (2005). Student processes. Adv. Appl. Prob. 37, 342365.Google Scholar
Hilfer, R. and Anton, L. (1995). Fractional master equations and fractal time random walks. Phys. Rev. E 51, R848R851.Google Scholar
Johnson, N. L., Kemp, A. W. and Kotz, S. (2005). Univariate Discrete Distributions, 3rd edn. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.Google Scholar
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (1998). Loss Models. From Data to Decisions. John Wiley, New York.Google Scholar
Kozubowski, T. J., Meerschaert, M. M. and Podgórski, K. (2006). Fractional Laplace motion. Adv. Appl. Prob. 38, 451464.Google Scholar
Kumar, A. and Vellaisamy, P. (2012). Fractional normal inverse Gaussian process. Methodology Comput. Appl. Prob. 14, 263283.CrossRefGoogle Scholar
Kumar, A., Meerschaert, M. M. and Vellaisamy, P. (2011). Fractional normal inverse Gaussian diffusion. Statist. Prob. Lett. 81, 146152.Google Scholar
Kumar, A., Nane, E. and Vellaisamy, P. (2011). Time-changed Poisson processes. Statist. Prob. Lett. 81, 18991910.Google Scholar
Linde, W. and Shi, Z. (2004). Evaluating the small deviation probabilities for subordinated Lévy processes. Stoch. Process. Appl. 113, 273287.Google Scholar
Mainardi, F., Gorenflo, R. and Scalas, E. (2004). A fractional generalization of the Poisson processes. Vietnam J. Math. 32, 5364.Google Scholar
Mainardi, F., Gorenflo, R. and Vivoli, A. (2005). Renewal processes of Mittag–Leffler and Wright type. Fract. Calc. Appl. Anal. 8, 738.Google Scholar
Meerschaert, M. M. and Scheffler, H. P. (2004). Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Prob. 41, 623638.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
Minkova, L. D. (2004). The Pólya–Aeppli process and ruin problems. J. Appl. Math. Stoch. Anal. 2004, 221234.Google Scholar
Orsingher, E. and Beghin, L. (2009). Fractional diffusion equations and processes with randomly varying time. Ann. Prob. 37, 206249.Google Scholar
Orsingher, E. and Polito, F. (2010). Fractional pure birth processes. Bernoulli 16, 858881.Google Scholar
Orsingher, E. and Polito, F. (2012). The space-fractional Poisson process. Statist. Prob. Lett. 82, 852858.Google Scholar
Patil, G. P. and Wani, J. K. (1965). On certain structural properties of the logarithmic series distribution and the first type Stirling distribution. Sankhyā A 27, 271280.Google Scholar
Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego, CA.Google Scholar
Saichev, A. I. and Zaslavsky, G. M. (1997). Fractional kinetic equations: solutions and applications. Chaos 7, 753764.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.Google Scholar
Sato, K. (2001). Subordination and self-decomposability. Statist. Prob. Lett. 54, 317324.Google Scholar
Scalas, E. (2012). A class of CTRWs: compound fractional Poisson processes. In: Fractional Dynamics, World Scientific, Hackensack, NJ, pp. 353374.Google Scholar