Journal of Applied Probability

From Hermite polynomials to multifractional processes

Renaud Marty

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider a class of multifractional processes related to Hermite polynomials. We show that these processes satisfy an invariance principle. To prove the main result of this paper, we use properties of the Hermite polynomials and the multiple Wiener integrals. Because of the multifractionality, we also need to deal with variations of the Hurst index by means of some uniform estimates.

Article information

Source
J. Appl. Probab., Volume 50, Number 2 (2013), 323-343.

Dates
First available in Project Euclid: 19 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1371648944

Digital Object Identifier
doi:10.1239/jap/1371648944

Mathematical Reviews number (MathSciNet)
MR3102483

Zentralblatt MATH identifier
1278.60066

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60G17: Sample path properties 60G22: Fractional processes, including fractional Brownian motion

Keywords
Multifractional process Hermite polynomial limit theorem sample path properties

Citation

Marty, Renaud. From Hermite polynomials to multifractional processes. J. Appl. Probab. 50 (2013), no. 2, 323--343. doi:10.1239/jap/1371648944. https://projecteuclid.org/euclid.jap/1371648944


Export citation

References

  • Benassi, A., Cohen, S. and Istas, J. (2002). Identification and properties of real harmonizable Lévy motions. Bernoulli 8, 97–115.
  • Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Math. Iberoamericana 13, 19–90.
  • Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.
  • Cohen, S. and Marty, R. (2008). Invariance principle, multifractional Gaussian processes and long-range dependence. Ann. Inst. H. Poincaré Prob. Statist. 44, 475–489.
  • Davydov, Y. A. (1970). The invariance principle for stationary processes. Theory Prob. Appl. 15, 487–498.
  • Dobrushin, R. L. (1979). Gaussian and their subordinated self-similar random generalized fields. Ann. Prob. 7, 1–28.
  • Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrscheinlichkeitsth. 50, 27–52.
  • Itô, K. (1951). Multiple Wiener integral. J. Math. Soc. Japan 3, 157–169.
  • Lacaux, C. and Marty, R. (2011). From invariance principles to a class of multifractional fields related to fractional sheets. Preprint. Available at http://hal.archives-ouvertes.fr/hal-00592188.
  • Marty, R. and Sølna, K. (2011). A general framework for waves in random media with long-range correlations. Ann. Appl. Prob. 21, 115–139.
  • Nelson, E. (1973). The free Markoff field. J. Funct. Anal. 12, 211–227.
  • Nourdin, I. and Peccati, G. (2012). Normal Approximations Using Malliavin Calculus (Cambridge Tracts Math. 192). Cambridge University Press.
  • Peltier, R. F. and Lévy Véhel, J. (1995). Multifractional Brownian motion: definition and preliminary results. Preprint. Available at http://hal.inria.fr/inria-00074045/.
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
  • Sly, A. (2007). Integrated fractional white noise as an alternative to multifractional Brownian motion. J. Appl. Prob. 44, 393–408.
  • Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287–302.
  • Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitsth. 50, 53–83.