Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-22T06:46:10.387Z Has data issue: false hasContentIssue false
  • English
  • Français

From Hermite Polynomials to Multifractional Processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Renaud Marty
Renaud Marty*
Affiliation:
Université de Lorraine
*
∗ Postal address: Institut Élie Cartan, Université de Lorraine, B.P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex, France. Email address: renaud.marty@univ-lorraine.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Multifractional processHermite polynomiallimit theoremsample path properties

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 60G17: Sample path properties 60G22: Fractional processes, including fractional Brownian motion
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 323 - 343
Copyright
© Applied Probability Trust 

References

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