Electronic Communications in Probability

A generalization of the space-fractional Poisson process and its connection to some Lévy processes

Federico Polito and Enrico Scalas

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The space-fractional Poisson process is a time-changed homogeneous Poisson process where the time change is an independent stable subordinator. In this paper, a further generalization is discussed that preserves the Lévy property. We introduce a generalized process by suitably time-changing a superposition of weighted space-fractional Poisson processes. This generalized process can be related to a specific subordinator for which it is possible to explicitly write the characterizing Lévy measure. Connections are highlighted to Prabhakar derivatives, specific convolution-type integral operators. Finally, we study the effect of introducing Prabhakar derivatives also in time.

Article information

Electron. Commun. Probab., Volume 21 (2016), paper no. 20, 14 pp.

Received: 24 June 2015
Accepted: 16 February 2016
First available in Project Euclid: 1 March 2016

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Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60G22: Fractional processes, including fractional Brownian motion 26A33: Fractional derivatives and integrals

fractional point processes Lévy processes Prabhakar integral Prabhakar derivative time-change subordination

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Polito, Federico; Scalas, Enrico. A generalization of the space-fractional Poisson process and its connection to some Lévy processes. Electron. Commun. Probab. 21 (2016), paper no. 20, 14 pp. doi:10.1214/16-ECP4383. https://projecteuclid.org/euclid.ecp/1456840981

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  • [1] B Baeumer and MM Meerschaert, Tempered stable Lévy motion and transient super-diffusion, Journal of Computational and Applied Mathematics 233 (2010), no. 10, 2438–2448.
  • [2] A-L Barabasi, Bursts: The hidden pattern behind everything we do, Dutton Adult, 2010.
  • [3] L Beghin and M D’Ovidio, Fractional Poisson process with random drift, Electron. J. Probab. 19 (2014), no. 122, 26.
  • [4] L Beghin and C Macci, Alternative forms of compound fractional Poisson processes, Abstract and Applied Analysis 2012 (2012).
  • [5] J Bertoin, Lévy processes, Cambridge University Press, 1998.
  • [6] NH Bingham, Limit theorems for occupation times of Markov processes, Probability Theory and Related Fields 17 (1971), no. 1, 1–22.
  • [7] SI Boyarchenko and S Levendorskii, Non-Gaussian Merton–Black–Scholes Theory, Advanced Series on Statistical Science and Applied Probability, vol. 9, World Scientific Singapore, 2002.
  • [8] H Câteau and A Reyes, Relation between single neuron and population spiking statistics and effects on network activity, Physical Review Letters 96 (2006), no. 5, 058101.
  • [9] E Çinlar and RA Agnew, On the Superposition of Point Processes, Journal of the Royal Statistical Society. Series B 30 (1968), no. 3, 576–581.
  • [10] LHY Chen and A Xia, Poisson process approximation for dependent superposition of point processes, Bernoulli 17 (2011), no. 2, 530–544.
  • [11] DR Cox and WL Smith, On the superposition of renewal processes, Biometrika 41 (1954), no. 1–2, 91–99.
  • [12] M Deger, M Helias, C Boucsein, and S Rotter, Statistical properties of superimposed stationary spike trains, Journal of Computational Neuroscience 32 (2012), no. 3, 443–463.
  • [13] L Devroye, A triptych of discrete distributions related to the stable law, Statistics & Probability Letters 18 (1993), 349–351.
  • [14] G Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer, Berlin, 1974.
  • [15] M D’Ovidio and F Polito, Fractional Diffusion-Telegraph Equations and their Associated Stochastic Solutions, [math.PR] arXiv:1307.1696 (2013).
  • [16] P Franken, A refinement of the Limit Theorem for the superposition of independent renewal processes, Teor. Veroyatnost. i Primen. 8 (1963), 320–328.
  • [17] R Garra, Gorenflo R, Polito F, and Tomovski Ž, Hilfer–Prabhakar derivatives and some applications, Applied Mathematics and Computation 242 (2014), 576–589.
  • [18] J Grandell, Doubly Stochastic Poisson Processes, Springer-Verlag, 1976.
  • [19] B Grigelionis, On the convergence of sums of random step processes to a Poisson process, Teor. Veroyatnost. i Primen. 8 (1963), 189–194.
  • [20] N Hohn and AN Burkitt, Shot noise in the leaky integrate-and-fire neuron, Physical Review E 63 (2001), 031902.
  • [21] Z-Q Jiang, W-J Xie, M-X Li, B Podobnik, W-X Zhou, and H E Stanley, Calling patterns in human communication dynamics, Proceedings of the National Academy of Sciences 110 (2013), 1600.
  • [22] AA Kilbas, M Saigo, and RK Saxena, Generalized Mittag–Leffler function and generalized fractional calculus operators, Integral Transforms and Special Functions 15 (2004), no. 1, 31–49.
  • [23] AA Kilbas, HM Srivastava, and JJ Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, 2006.
  • [24] JFC Kingman, On doubly stochastic Poisson processes, Proc. Camb. Phil. Soc, vol. 60, Cambridge University Press, 1964, p. 923.
  • [25] I Koponen, Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process, Physical Review E 52 (1995), no. 1, 1197.
  • [26] AE Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer, 2007.
  • [27] AN Lageras, A renewal-process-type expression for the moments of inverse subordinators, Journal of Applied Probability 42 (2005), 1134–1144.
  • [28] N Laskin, Fractional poisson process, Communications in Nonlinear Science and Numerical Simulation 8 (2003), no. 3–4, 201–213.
  • [29] B Lindner, Superposition of many independent spike trains is generally not a Poisson process, Physical Review E 73 (2006), 022901.
  • [30] F Mainardi, R Gorenflo, and E Scalas, A fractional generalization of the poisson process, Vietnam Journal of Mathematics 32 SI (2004), 65–75.
  • [31] MM Meerschaert and H-P Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times, Journal of Applied Probability 41 (2004), no. 3, 623–638.
  • [32] MM Meerschaert and A Sikorskii, Stochastic Models for Fractional Calculus, vol. 43, de Gruyter, 2011.
  • [33] E Orsingher and F Polito, The space-fractional Poisson process, Statistics & Probability Letters 82 (2012), no. 4, 852–858.
  • [34] E Orsingher and B Toaldo, Counting processes with Bernštein intertimes and random jumps, Journal of Applied Probability 52 (2015), no. 4, 1028–1044.
  • [35] I Podlubny, Fractional Differential Equations, vol. 198, Academic press, 1998.
  • [36] M Politi, T Kaizoji, and E Scalas, Full characterization of the fractional poisson process, Europhysics Letters 96 (2011), no. 2, 20004.
  • [37] J Rosiński, Tempering stable processes, Stochastic processes and their applications 117 (2007), no. 6, 677–707.
  • [38] T Shimokawa, A Rogel, K Pakdaman, and S Sato, Stochastic resonance and spike-timing precision in an ensemble of leaky integrate and fire neuron models, Physical Review E 59 (1999), 3461.
  • [39] FW Steutel and K van Harn, Discrete Analogues of Self-Decomposability and Stability, The Annals of Probability 7 (1979), no. 5, 893–899.
  • [40] M Veillette and MS Taqqu, Using differential equations to obtain joint moments of first-passage times of increasing Lévy processes, Statistics & Probability Letters 80 (2010), 697–705.