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Limit theorems for the fractional nonhomogeneous Poisson process

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  12 July 2019

Nikolai Leonenko ,
Enrico Scalas  and
Mailan Trinh
Nikolai Leonenko*
Affiliation:
Cardiff University
Enrico Scalas*
Affiliation:
University of Sussex
Mailan Trinh*
Affiliation:
University of Sussex
*
*Postal address: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK.
**Postal address: Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, BN1 9QH, UK.
**Postal address: Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, BN1 9QH, UK.
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Abstract

The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

Keywords

Fractional point processeslimit theoremPoisson processadditive processLévy processtime changesubordination

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F05: Central limit and other weak theorems 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 246 - 264
Copyright
© Applied Probability Trust 2019 

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References

Aletti, G., Leonenko, N. and Merzbach, E. (2018). Fractional Poisson fields and martingales. J. Statist. Phys. 170, 700–730.CrossRefGoogle Scholar
Anscombe, F. J. (1952). Large-sample theory of sequential estimation. Proc. Camb. Phil. Soc. 48, 600–607.CrossRefGoogle Scholar
Becker-Kern, P., Meerschaert, M. M. and Scheffler, H.-P. (2004). Limit theorems for coupled continuous time random walks. Ann. Prob. 32, 730–756.Google Scholar
Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electron. J. Prob. 14, 1790–1827.CrossRefGoogle Scholar
Beghin, L. and Orsingher, E. (2010). Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Prob. 15, 684–709.CrossRefGoogle Scholar
Biard, R. and Saussereau, B. (2014). Fractional Poisson process: long-range dependence and applications in ruin theory. J. Appl. Prob. 51, 727–740.CrossRefGoogle Scholar
Biard, R. and Saussereau, B. (2016). Correction: “Fractional Poisson process: long-range dependence and applications in ruin theory”. J. Appl. Prob. 53, 1271–1272.CrossRefGoogle Scholar
Bielecki, T. R. and Rutkowski, M. (2002). Credit risk: modelling, valuation and hedging. Springer Finance. Springer, Berlin.Google Scholar
Bingham, N. H. (1971). Limit theorems for occupation times of Markov processes. Z. Wahrscheinlichkeitsth. 14, 1–22.CrossRefGoogle Scholar
Brémaud, P. (1981). Point Processes and Queues. Springer, New York.CrossRefGoogle Scholar
Cahoy, D. O., Polito, F. and Phoha, V. (2015). Transient behavior of fractional queues and related processes. Methodol. Comput. Appl. Prob. 17, 739–759.CrossRefGoogle Scholar
Cahoy, D. O., Uchaikin, V. V. and Woyczynski, W. A. (2010). Parameter estimation for fractional Poisson processes. J. Statist. Plann. Infer. 140, 3106–3120.CrossRefGoogle Scholar
Cohen, A. M. (2007). Numerical Methods for Laplace Transform Inversion (Numerical Methods Alg. 5). Springer, New York.Google Scholar
Cont, R. and Tankov, P. (2004). Financial modelling with jump processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Cox, D. R. (1955). Some statistical methods connected with series of events. J. R. Statist. Soc. Ser. B. 17, 129–157; discussion, 157–164.Google Scholar
Csörgő, M. and Fischler, R. (1973). Some examples and results in the theory of mixing and random-sum central limit theorems. Collection of articles dedicated to the memory of Alfréd 402 Rényi, II. Period. Math. Hungar. 3, 41–57.CrossRefGoogle Scholar
Daley, D. J. and Vere-jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.Google Scholar
Di Crescenzo, A., Martinucci, B. and Meoli, A. (2016). A fractional counting process and its connection with the Poisson process. ALEA Lat. Am. J. Prob. Math. Statist. 13, 291–307.Google Scholar
Ehrenberg, A. S. (1988). Repeat-buying: Facts, Theory and Data. Oxford University Press, New York.Google Scholar
Embrechts, P. and Hofert, M. (2013). A note on generalized inverses. Math. Methods Oper. Res. 77, 423–432.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Fulger, D., Scalas, E. and Germano, G. (2008). Monte carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Physical Review E 77, 021122.CrossRefGoogle ScholarPubMed
Gergely, T. and Yezhow, I. I. (1973). On a construction of ordinary Poisson processes and their modelling. Z. Wahrscheinlichkeitsth. 27, 215–232.CrossRefGoogle Scholar
Gergely, T. and Yezhow, I. I. (1975). Asymptotic behaviour of stochastic processes modelling an ordinary Poisson process. Period. Math. Hungar. 6, 203–211.CrossRefGoogle Scholar
Germano, G., Politi, M., Scalas, E. and Schilling, R. L. (2009). Stochastic calculus for uncoupled continuous-time random walks. Phys. Rev. E 79, 066102.CrossRefGoogle ScholarPubMed
Gnedenko, B. V. and Kovalenko, I. N. (1968). Introduction to Queueing Theory. Translated from Russian by Kondor, R.. Daniel Davey, Hartford, Conn.Google Scholar
Grandell, J. (1976). Doubly Stochastic Poisson Processes (Lecture Notes Math. 529). Springer, Berlin.CrossRefGoogle Scholar
Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.CrossRefGoogle Scholar
Gut, A. (1974). On the moments and limit distributions of some first passage times. Ann. Prob. 2, 277–308.CrossRefGoogle Scholar
Gut, A. (2013). Probability: A Graduate Course, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes (Fundamental Principles Math. Sci. 288), 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Khintchine, A. Y. (1969). Mathematical Methods in the Theory of Queueing (Griffin’s Statist. Monogr. Courses 7), 2nd edn. Hafner Publishing, New York.Google Scholar
Kingman, J. (1964). On the doubly stochastic Poisson processes. Proc. Camb. Phil. Soc. 60, 923–930.CrossRefGoogle Scholar
Kozubowski, T. J. and Rachev, S. T. (1999). Univariate geometric stable laws. J. Comput. Anal. Appl. 1, 177–217.Google Scholar
Laskin, N. (2003). Fractional Poisson process. Commun. Nonlinear Sci. Numer. Simul. 8, 201–213.CrossRefGoogle Scholar
Leonenko, N. and Merzbach, E. (2015). Fractional Poisson fields. Methodol. Comput. Appl. Prob. 17, 155–168.CrossRefGoogle Scholar
Leonenko, N., Scalas, E. and Trinh, M. (2017). The fractional non-homogeneous Poisson process. Statist. Prob. Lett. 120, 147–156.CrossRefGoogle Scholar
Maheshwari, A. and Vellaisamy, P. (2016). On the long-range dependence of fractional Poisson and negative binomial processes. J. Appl. Prob. 53, 989–1000.CrossRefGoogle Scholar
Mainardi, F., Gorenflo, R. and Scalas, E. (2004). A fractional generalization of the Poisson processes. Vietnam J. Math. 32, 53–64.Google Scholar
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
Meerschaert, M. M. and Scheffler, H.-P. (2004). Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Prob. 41, 623–638.CrossRefGoogle Scholar
Meerschaert, M. M. and Sikorskii, A. (2012). Stochastic Models for Fractional Calculus (De Gruyter Stud. Math. 43). Walter de Gruyter, Berlin.Google Scholar
Meerschaert, M. M. and Straka, P. (2013). Inverse stable subordinators. Math. Model. Nat. Phenom. 8, 1–16.CrossRefGoogle ScholarPubMed
Politi, M., Kaizoji, T. and Scalas, E. (2011). Full characterisation of the fractional poisson process. Europhysics Letters 96.CrossRefGoogle Scholar
Post, E. L. (1930). Generalized differentiation. Trans. Amer. Math. Soc. 32, 723–781.CrossRefGoogle Scholar
Richter, W. (1965). Übertragung von Grenzaussagen für Folgen von zufälligen Grössen auf Folgen mit zufälligen Indizes. Teor. Veroyat. Primen. 10, 82–94. English translation: Theoret. Prob. Appl. 10, 74–84.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.Google Scholar
Serfozo, R. F. (1972a). Conditional Poisson processes. J. Appl. Prob. 9, 288–302.CrossRefGoogle Scholar
Serfozo, R. F. (1972b). Processes with conditional stationary independent increments. J. Appl. Prob. 9, 303–315.CrossRefGoogle Scholar
Watanabe, S. (1964). On discontinuous additive functionals and Lévy measures of a Markov process. Japan. J. Math. 34, 53–70.CrossRefGoogle Scholar
Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.CrossRefGoogle Scholar
Widder, D. V. (1941). The Laplace Transform (Princeton Math. Ser. 6). Princeton University Press.Google Scholar