Advertisement

Limit Theorems for Quadratic Variations of the Lei–Nualart Process

  • Salwa Bajja
  • Khalifa Es-Sebaiy
  • Lauri Viitasaari
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 271)

Abstract

Let X be a Lei–Nualart process with Hurst index \(H\in (0, 1)\), \(Z_{1}\) be an Hermite random variable. For any \(n \ge 1\), set
$$V_{n}=\sum _{k=0}^{n-1}\left[ n^{2H}(\varDelta _k X)^{2}-n^{2H}\mathrm {I\! E}(\varDelta _k X)^{2}\right] .$$
The aim of the current paper is to derive, in the case when the Hurst index verifies \(H > 3/4\), an upper bound for the total variation distance between the laws \(\mathcal {L}(Z_n)\) and \(\mathcal {L}(Z_{1})\), where \(Z_{n}\) stands for the correct renormalization of \(V_{n}\) which converges in distribution towards \(Z_{1}\). We derive also the asymptotic behavior of quadratic variations of process X in the critical case \(H=3/4\), i.e. an upper bound for the total variation distance between the \(\mathcal {L}(Z_n)\) and the Normal law.

Keywords

Hermite random variable Gaussian analysis Malliavin calculus Convergence in law Berry–Esseen bounds 

References

  1. 1.
    Baxter, G.: A strong limit theorem for Gaussian processes. Proc. Am. Soc. 7, 522–527 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Begyn, A.: Quadratic variations along irregular subdivisions for Gaussian processes. Electron. J. Probab. 10(20), 691–717 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Begyn, A.: Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. Bernoulli 13(3), 712–753 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Begyn, A.: Functional limit theorems for generalized quadratic variations of Gaussian processes. Stoch. Process. Appl. 117, 1848–1869 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Benassi, A., Cohen, S., Istas, J.: Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett. 39, 337–345 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Benassi, A., Cohen, S., Istas, J., Jaffard, S.: Identification of filtered white noises. Stoch. Probab. Appl. 75, 31–49 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bercu, B., Nourdin, I., Taqqu, M.S.: Almost sure central limit theorems on the Wiener space. Stoch. Process. 120, 1607–1628 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Breton, J.C., Nourdin, I.: Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion. Electron. Commun. Probab. 13, 482–493 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Breuer, P., Major, P.: Central limit theorems for non-linear functionals of Gaussian fields. J. Multivar. Anal. 13, 425–441 (1983)CrossRefGoogle Scholar
  10. 10.
    Coeurjolly, J.: Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4, 199–227 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Coeurjolly, J.: Identification of the multifractional Brownian motion. Bernoulli 11, 987–1009 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cohen, S., Guyon, X., Perrin, O., Pontier, M.: Singularity functions for fractional processes: application to fractional Brownian sheet. Annales de l’Institut Henri Poincare 42(2), 187–205 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    De La Vega, W.F.: On almost sure convergence of quadratic Brownian variation. Ann. Probab. 2, 551–552 (1974)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dobrushin, R.L., Major, P.: Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrsch. verw. Gebiete 50, 27–52 (1979)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Dudley, R.M.: Sample functions of the Gaussian process. Ann. Probab. 1, 66–103 (1973)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Giraitis, L., Surgailis, D.: CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. verw. Gebiete 70, 191–212 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gladyshev, E.G.: A new limit theorem for stochastic processes with Gaussian increments. Theory Probab. Appl. 6(1), 52–61 (1961)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Guyon, X., Leon, J.: Convergence en loi des h-variations d’un processus Gaussien stationnaire sur R. Annales de l’Institut Henri Poincare 25(3), 265–282 (1989)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Harnett, D., Nualart, D.: Decomposition and limit theorems for a class of self-similar Gaussian processes (2015). Submitted. arXiv:1508.06641
  20. 20.
    Istas, J., Lang, G.: Quadratic variations and estimation of the local holder index of a Gaussian process. Annal. de l’Institut Henri Poincare 33(4), 407–436 (1997)CrossRefGoogle Scholar
  21. 21.
    Klein, R., Gine, E.: On quadratic variations of processes with Gaussian increments. Ann. Probab. 3(4), 716–721 (1975)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kubilius, K., Melichov, D.: On the convergence rates of Gladyshev’s Hurst index estimator. Nonlinear Anal. Model. Control 15, 445–450 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lacaux, C.: Real harmonizable multifractional Lévy motions. Annales de l’Institut Henri Poincare 40(3), 259–277 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lei, P., Nualart, D.: A decomposition of the bi-fractional Brownian motion and some applications. Stat. Probab. Lett. 79, 619–624 (2009)CrossRefGoogle Scholar
  25. 25.
    Levental, S., Erickson, R.V.: On almost sure convergence of the quadratic variation of Brownian motion. Stoch. Process. Appl. 106(2), 317–333 (2003)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Malukas, R.: Limit theorems for a quadratic variation of Gaussian processes. Nonlinear Anal. Model. Control 16(4), 435–452 (2011)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Norvaisa, R.: A complement to Gladyshevs theorem. Lith. Math. J. 51, 26–35 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Nourdin, I., Peccati, G.: Stein’s method on Wiener chaos. Probab. Theory Relat. Fields 145, 1–2 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  30. 30.
    Peccati, G., Taqqu, M.: Wiener Chaos: Moments, Cumulants and Diagrams. A Survey with Computer Implementation. Bocconi University Press, Springer, Berlin (2011)CrossRefGoogle Scholar
  31. 31.
    Perrin, O.: Quadratic variation for Gaussian processes and application to time deformation. Stoch. Process. Appl. 40(3), 293–305 (1999)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Ruiz de Chavez, J., Tudor, C.: A decomposition of subfractional Brownian motion. Math. Rep. 61, 67–74 (2009)Google Scholar
  33. 33.
    Taqqu, M.: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Probab. Theory Relat. Fields 31(4), 287–302 (1975)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Taqqu, M.: Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50, 53–83 (1979)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Viitasaari, L.: Sufficient and necessary conditions for limit theorems for quadratic variations of Gaussian sequences (2015). Submitted. arXiv:1502.01370

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Salwa Bajja
    • 1
  • Khalifa Es-Sebaiy
    • 2
  • Lauri Viitasaari
    • 3
  1. 1.National School of Applied Sciences—MarrakeshCadi Ayyad UniversityMarrakeshMorocco
  2. 2.Department of Mathematics, Faculty of ScienceKuwait UniversityKuwaitKuwait
  3. 3.Department of Mathematics and System AnalysisAalto University School of ScienceAaltoFinland

Personalised recommendations