Rosenblatt Laplace Motion

  • Oana Lupaşcu-Stamate
  • Ciprian A. TudorEmail author


Motivated by several works on the modelization of hydraulic conductivity, we introduce the Rosenblatt Laplace motion, by subordinating the Rosenblatt process to an independent Gamma process. We derive the basic properties of this new fractal-type stochastic process and we also make a numerical analysis of it. In particular, we compute numerically its moments and cumulants and we provide a method to simulate its sample paths.


fractional Laplace motion Rosenblatt process fractional Brownian motion cumulants subordination Gamma process hydraulic conductivity 

Mathematics Subject Classification

60G07 60G18 60E07 



  1. 1.
    Abry, P., Pipiras, V.: Wavelet-based synthesis of the Rosenblatt process. Signal Process. 86, 2326–2339 (2006)zbMATHCrossRefGoogle Scholar
  2. 2.
    Ahn, V., Leonenko, N., Olenko, A.: On the rate of convergence to Rosenblatt-type distribution. J. Math. Anal. Appl. 425(1), 111–132 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Araya, H., Tudor, C.A.: Behavior of the Hermite sheet with respect to the Hurst index. Stoch. Process. Appl. Preprint (2018).
  4. 4.
    Bardet, J.-M.: Software: Rosenblatt processes from wavelet based procedure (2008).
  5. 5.
    Bochner, S.: Subordination of non-Gaussian stochastic processes. Proc. Natl. Acad. Sci. USA 48(1), 19–22 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dobrushin, R.L., Major, P.: Non-central limit theorems for non-linear functionals of Gaussian fields Gaussian field. Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 27–52 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Fauth, A., Tudor, C.A.: Handbook of high-frequency trading and modeling in finance. In: Florescu, I., Mariani, M.C., Stanley, H.E., Viens, F.G. (eds.) Multifractal Random Walk Driven by a Hermite Process, pp. 221–250. Wiley, New York (2016)Google Scholar
  8. 8.
    Gajda, J., Wylomanska, A., Kumar, A.: Generalized fractional Laplace motion. Stat. Probab. Lett. 124, 101–109 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Heyde, C.C.: A risky asset model with strong dependence through fractal activity time. J. Appl. Probab. 36(4), 1234–1239 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kozubowski, T., Meerschaert, M., Podgorski, K.: Fractional Laplace motion. Adv. Appl. Probab. 38, 451–464 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kumar, A., Wylomanska, A., Poloczanski, R., Sundar, S.: Fractional Brownian motion time-changed by gamma and inverse gamma process. Physica A 468, 648–667 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lin, G.D.: Recent developments on the moment problem. Preprint (2017).
  13. 13.
    Lupaşcu, O.: Subordination in the sense of Bochner of \(L^p\) semigroups and associated Markov processes. Acta Math. Sin. Engl. Ser. 30, 187–196 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Maejima, M., Tudor, C.A.: On the distribution of the Rosenblatt process. Stat. Probab. Lett. 83(6), 1490–1495 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Meerschaert, M., Kozubowski, T., Molz, F., Lu, S.: Fractional Laplace motion for hydraulic conductivity. Geophys. Res. Lett. 31, L08501 (2004)CrossRefGoogle Scholar
  16. 16.
    Nourdin, I.: Selected Aspects of the Fractional Brownian Motion. Springer-Bocconi, Berlin (2012)zbMATHCrossRefGoogle Scholar
  17. 17.
    Nourdin, I., Peccati, G.: Normal Approximations with Malliavin Calculus From Stein ’s Method to Universality. Cambridge University Press, Cambridge (2012)zbMATHCrossRefGoogle Scholar
  18. 18.
    Nualart, D.: The Malliavin Calculus and Related Topics. Probability and Its Applications, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  19. 19.
    Pipiras, V., Taqqu, M.: Long-Range Dependence and Self-Similarity. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2017)zbMATHCrossRefGoogle Scholar
  20. 20.
    Sato, K.-I.: Levy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  21. 21.
    Schilling, R., Song, R., Vondracek, Z.: Bernstein Functions, Theory and Applications. In: Bernstein functions, 2 edn, de Gruyter Studies in Mathematics, vol. 37 (2012)Google Scholar
  22. 22.
    Shigekawa, I.: Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ. 20(2), 263–289 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Taqqu, M.: Weak convergence to the fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31, 287–302 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Torres, S., Tudor, C.A.: Donsker type theorem for the Rosenblatt process and a binary market model. Stoch. Anal. Appl. 27, 555–573 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Tudor, C.A.: Analysis of Variations for Self-Similar Processes. A Stochastic Calculus Approach. Probability and Its Applications (New York). Springer, Cham (2013)Google Scholar
  26. 26.
    Veillette, M.S., Taqqu, M.S.: Berry–Esseen and edgeworth approximations for the normalized tail of an infinite sum of independent weighted gamma random variables. Stoch. Process. Appl. 122, 885–909 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Veillette, M.S., Taqqu, M.S.: Properties and numerical evaluation of the Rosenblatt process. Bernoulli 19(3), 982–1005 (2013)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Mathematical Statistics and Applied Mathematics of the Romanian AcademyBucharestRomania
  2. 2.Laboratoire Paul PainlevéUniversité de Lille 1Villeneuve d’AscqFrance

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