Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Invariance principle, multifractional Gaussian processes and long-range dependence

Serge Cohen and Renaud Marty

Full-text: Open access

Abstract

This paper is devoted to establish an invariance principle where the limit process is a multifractional Gaussian process with a multifractional function which takes its values in (1/2, 1). Some properties, such as regularity and local self-similarity of this process are studied. Moreover the limit process is compared to the multifractional Brownian motion.

Résumé

Ce papier a pour but d’établir un principe d’invariance dont le processus limite est gaussien et multifractionnaire avec une fonction de Hurst à valeurs dans (1/2, 1). Des propriétés telles que la régularité et l’autosimilarité locale de ce processus sont étudiées. De plus, le processus limite est comparé au mouvement brownien multifractionnaire.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 44, Number 3 (2008), 475-489.

Dates
First available in Project Euclid: 26 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1211819421

Digital Object Identifier
doi:10.1214/07-AIHP127

Mathematical Reviews number (MathSciNet)
MR2451054

Zentralblatt MATH identifier
1176.60021

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60G15: Gaussian processes

Keywords
Invariance principle Long range dependence Multifractional process Gaussian processes

Citation

Cohen, Serge; Marty, Renaud. Invariance principle, multifractional Gaussian processes and long-range dependence. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 3, 475--489. doi:10.1214/07-AIHP127. https://projecteuclid.org/euclid.aihp/1211819421


Export citation

References

  • R. J. Adler. The Geometry of Random Fields. Wiley, London, 1981.
  • A. Ayache, S. Cohen and J. Lévy-Véhel. The covariance structure of multifractional Brownian motion, with application to long range dependence, Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2000.
  • A. Ayache and J. Lévy-Véhel. The generalized multifractional Brownian motion. Stat. Inference for Stoch. Process. 3 (2000) 7–18.
  • A. Benassi, S. Cohen and J. Istas. Identifying the multifractional function of a Gaussian process. Statist. Probab. Lett. 39 (1998) 337–345.
  • A. Benassi, S. Cohen and J. Istas. Identification and properties of real harmonizable Lévy motions. Bernoulli 8 (2002) 97–115.
  • A. Benassi, S. Jaffard and D. Roux. Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 (1997) 19–90.
  • P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968.
  • S. Cohen. From self-similarity to local self-similarity: the estimation problem. In Fractal in Engineering 3–16. J. Lévy-Véhel and C. Tricot (Eds). Springer, London, 1999.
  • Y. Davydov. The invariance principle for stationary processes. Theory Probab. Appl. 15 (1970) 487-498.
  • M. Dekking, J. Lévy-Véhel, E. Lutton and C. Tricot. Fractals: Theory and Applications in Engineering. Springer, London, 1999.
  • C. Lacaux. Real Harmonizable multifractional Lévy motions. Ann. Inst. H. Poincaré, Probab. Statist. 40 (2004) 259–277.
  • B. Mandelbrot, J. V. Ness. Fractional Brownian motion, fractional noises and applications. SIAM Review 10 (1968) 422–437.
  • R. Peltier and J. Lévy-Véhel. Multifractional Brownian motion: definition and preliminary results. INRIA research report, RR-2645, 1995.
  • A. Philippe, D. Surgailis and M.-C. Viano. Time-varying fractionally integrated processes with nonstationary long memory, Theory Probab. Appl. (2007). To appear.
  • G. Samorodnitsky and M. S. Taqqu. Stable non-Gaussian Random Processes. Chapman and Hall, New York, 1994.