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Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse

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Abstract

In this paper, we study the fractional Poisson process (FPP) time-changed by an independent Lévy subordinator and the inverse of the Lévy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property, and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. The bivariate distributions of the TCFPP-II are derived. Some specific examples for both the processes are discussed. Finally, we present simulations of the sample paths of these processes.

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References

  1. Aletti, G., Leonenko, N., Merzbach, E.: Fractional Poisson fields and martingales. (2016) arXiv:1601.08136

  2. Alrawashdeh, M.S., Kelly, J.F., Meerschaert, M.M., Scheffler, H.-P.: Applications of inverse tempered stable subordinators. Comput. Math. Appl. 73, 892–905 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  4. Avramidis, A.N., L’ecuyer, P., Tremblay, P.-A.: Efficient simulation of gamma and variance-gamma processes. In: Simulation Conference, 2003. Proceedings of the 2003 Winter 1, pp. 319–326 (2003)

  5. Beghin, L., Orsingher, E.: Fractional Poisson processes and related planar random motions. Electron. J. Probab. 14, 1790–1827 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beghin, L., Orsingher, E.: Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Probab. 15, 684–709 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beghin, L., Macci, C.: Fractional discrete processes: compound and mixed Poisson representations. J. Appl. Probab. 51, 9–36 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beghin, L., Vellaisamy, P.: Space-fractional versions of the negative binomial and Polya-type processes. Comput. Appl. Probab. Methodol. (2017). https://doi.org/10.1007/s11009-017-9561-8

  9. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  10. Bingham, N.H.: Limit theorems for occupation times of Markov processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17, 1–22 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  11. Buchak, K., Sakhno, L.: Compositions of Poisson and gamma processes. Mod. Stoch. Theory Appl. 4, 161–188 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cahoy, D.O., Uchaikin, V.V., Woyczynski, W.A.: Parameter estimation for fractional Poisson processes. J. Stat. Plan. Inference 140, 3106–3120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton (2004)

    MATH  Google Scholar 

  14. Cox, D.R., Lewis, P.A.W.: The Statistical Analysis of Series of Events. Wiley, New York (1966)

    Book  MATH  Google Scholar 

  15. D’Ovidio, M., Nane, E.: Time dependent random fields on spherical non-homogeneous surfaces. Stoch. Process. Appl. 124, 2098–2131 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hofert, M.: Sampling exponentially tilted stable distributions. ACM Trans. Model. Comput. Simul. 22, 3 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jørgensen, B.: Statistical properties of the generalized inverse Gaussian distribution. In: Lecture Notes in Statistics, vol. 9. Springer, New York (1982)

  18. Jumarie, G.: Fractional master equation: non-standard analysis and Liouville–Riemann derivative. Chaos Solitons Fractals 12, 2577–2587 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kanter, M.: Stable densities under change of scale and total variation inequalities. Ann. Probab. 3, 697–707 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kumar, A., Nane, E., Vellaisamy, P.: Time-changed Poisson processes. Stat. Probab. Lett. 81, 1899–1910 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kumar, A., Vellaisamy, P.: Inverse tempered stable subordinators. Stat. Probab. Lett. 103, 134–141 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kumar, A., Wyłomańska, A., Połoczański, R., Sundar, S.: Fractional Brownian motion time-changed by gamma and inverse gamma process. Physica A 468, 648–667 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Laskin, N.: Fractional Poisson process. Commun. Nonlinear Sci. Numer. Simul. 8, 201–213 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Leonenko, N.N., Meerschaert, M.M., Schilling, R.L., Sikorskii, A.: Correlation structure of time-changed Lévy processes. Commun. Appl. Ind. Math. 6, e-483 (2014)

    MATH  Google Scholar 

  25. Leonenko, N.N., Merzbach, E.: Fractional Poisson fields. Methodol. Comput. Appl. 17, 155–168 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Leonenko, N.N., Scalas, E., Trinh, M.: The fractional non-homogeneous Poisson process. Stat. Probab. Lett. 120, 147–156 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Maheshwari, A., Vellaisamy, P.: Non-homogeneous space-time fractional Poisson processes. arXiv:1607.06016 [math.PR] (2017)

  28. Maheshwari, A., Vellaisamy, P.: On the long-range dependence of fractional Poisson and negative binomial processes. J. Appl. Probab. 53, 989–1000 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mainardi, F., Gorenflo, R., Scalas, E.: A fractional generalization of the Poisson processes. Vietnam J. Math. 32, 53–64 (2004)

    MathSciNet  MATH  Google Scholar 

  30. Meerschaert, M.M., Nane, E., Vellaisamy, P.: The fractional Poisson process and the inverse stable subordinator. Electron. J. Probab. 16, 1600–1620 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Meerschaert, M.M., Scheffler, H.-P.: Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41, 623–638 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mittag-Leffler, G.M.: Sur la nouvelle fonction \(E_\alpha (x)\). C. R. Acad. Sci. Paris 137, 554–558 (1903)

    MATH  Google Scholar 

  33. Orsingher, E., Polito, F.: The space-fractional Poisson process. Stat. Probab. Lett. 82, 852–858 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Orsingher, E., Polito, F.: On the integral of fractional Poisson processes. Stat. Probab. Lett. 83, 1006–1017 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Orsingher, E., Toaldo, B.: Counting processes with Bernštein intertimes and random jumps. J. Appl. Probab. 52, 1028–1044 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Prabhakar, T.R.: A singular integral equation with a generalized Mittag Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)

    MathSciNet  MATH  Google Scholar 

  37. Repin, O.R., Saichev, A.I.: Fractional Poisson law. Radiophys. Quantum Electron. 43, 738–741 (2000)

    Article  MathSciNet  Google Scholar 

  38. Schilling, R.L., Song, R., Vondraček, Z.: Bernstein functions. In: de Gruyter Studies in Mathematics, vol. 37, 2nd edn. Walter de Gruyter & Co., Berlin (2012)

  39. Veillette, M., Taqqu, M.S.: Numerical computation of first passage times of increasing Lévy processes. Methodol. Comput. Appl. Probab. 12, 695–729 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Vellaisamy, P., Kumar, A.: First-exit times of an inverse Gaussian process. To appear in Stochastics (2017) arXiv:1105.1468

  41. Vellaisamy, P., Maheshwari, A.: Fractional negative binomial and Polya processes. To appear in Probab. Math. Statist. (2017)

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Acknowledgements

A part of this work was done while the second author was visiting the Department of Statistics and Probability, Michigan State University, during Summer 2016. The authors thank the referee for some useful comments.

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Authors and Affiliations

  1. Operations Management and Quantitative Techniques Area, Indian Institute of Management Indore, Prabandh Shikhar, Rau-Pithampur Road, Indore, Madhya Pradesh, 453556, India

    A. Maheshwari

  2. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

    P. Vellaisamy

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  1. A. Maheshwari
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  2. P. Vellaisamy
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Correspondence to A. Maheshwari.

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Maheshwari, A., Vellaisamy, P. Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse. J Theor Probab 32, 1278–1305 (2019). https://doi.org/10.1007/s10959-017-0797-6

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  • DOI: https://doi.org/10.1007/s10959-017-0797-6

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