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On the long-range dependence of fractional Poisson and negative binomial processes

Part of: Stochastic processes

Published online by Cambridge University Press:  09 December 2016

A. Maheshwari  and
P. Vellaisamy
A. Maheshwari*
Affiliation:
Indian Institute of Technology Bombay
P. Vellaisamy*
Affiliation:
Indian Institute of Technology Bombay
*
* Postal address: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India.
* Postal address: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India.
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Abstract

We discuss the short-range dependence (SRD) property of the increments of the fractional Poisson process, called the fractional Poissonian noise. We also establish that the fractional negative binomial process (FNBP) has the long-range dependence (LRD) property, while the increments of the FNBP have the SRD property. Our definitions of the SRD/LRD properties are similar to those for a stationary process and different from those recently used in Biard and Saussereau (2014).

Keywords

Long-range dependencefractional Poisson processfractional negative binomial process

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 989 - 1000
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Beghin, L. and Orsingher, E. (2009).Fractional Poisson processes and related planar random motions.Electron. J. Prob. 14,1790–1827.CrossRefGoogle Scholar
[2] Biard, R. and Saussereau, B. (2014).Fractional Poisson process: long-range dependence and applications in ruin theory.J. Appl. Prob. 51,727–740.Google Scholar
[3] Ding, Z.,Granger, C. W. J. and Engle, R. F. (1993).A long memory property of stock market returns and a new model.J. Empirical Finance 1,83–106.Google Scholar
[4] Doukhan, P.,Oppenheim, G. and Taqqu, M. S. (eds) (2003).Theory and Applications of Long-Range Dependence.Birkhäuser,Boston, MA.Google Scholar
[5] D'Ovidio, M. and Nane, E. (2014).Time dependent random fields on spherical non-homogeneous surfaces.Stoch. Process. Appl. 124,2098–2131.CrossRefGoogle Scholar
[6] Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II,2nd edn.John Wiley,New York.Google Scholar
[7] Heyde, C. C. and Yang, Y. (1997).On defining long-range dependence.J. Appl. Prob. 34,939–944.Google Scholar
[8] Karagiannis, T.,Molle, M. and Faloutsos, M. (2004).Long-range dependence ten years of internet traffic modeling.IEEE Internet Comput. 8,57–64.CrossRefGoogle Scholar
[9] Kozubowski, T. J. and Podgórski, K. (2009).Distributional properties of the negative binomial Lévy process.Prob. Math. Statist. 29,43–71.Google Scholar
[10] Laskin, N. (2003).Fractional Poisson process.Commun. Nonlinear Sci. Numer. Simul. 8,201–213.CrossRefGoogle Scholar
[11] Leonenko, N. N.,Meerschaert, M. M.,Schilling, R. L. and Sikorskii, A. (2014).Correlation structure of time-changed Lévy processes.Commun. Appl. Ind. Math. 6,e-483.Google Scholar
[12] Mainardi, F.,Gorenflo, R. and Scalas, E. (2004).A fractional generalization of the Poisson processes.Vietnam J. Math. 32,53–64.Google Scholar
[13] 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
[14] Pagan, A. (1996).The econometrics of financial markets.J. Empirical Finance 3,15–102.CrossRefGoogle Scholar
[15] Varotsos, C. and Kirk-Davidoff, D. (2006).Long-memory processes in ozone and temperature variations at the region 60°s-60°n .Atmos. Chem. Phys. 6,4093–4100.CrossRefGoogle Scholar
[16] Vellaisamy, P. and Maheshwari, A. (2016).Fractional negative binomial and Pólya processes. To appear in Prob. Math. Statist. Google Scholar