Invariance principle, multifractional gaussian processes and long-range dependence

Serge Cohen; Renaud Marty

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 3, page 475-489
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Cohen, Serge, and Marty, Renaud. "Invariance principle, multifractional gaussian processes and long-range dependence." Annales de l'I.H.P. Probabilités et statistiques 44.3 (2008): 475-489. <http://eudml.org/doc/77979>.

@article{Cohen2008,
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.},
author = {Cohen, Serge, Marty, Renaud},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {invariance principle; long range dependence; multifractional process; gaussian processes; centered Gaussian field; finite-dimensional convergence; multifractional Gaussian process; long-range dependence},
language = {eng},
number = {3},
pages = {475-489},
publisher = {Gauthier-Villars},
title = {Invariance principle, multifractional gaussian processes and long-range dependence},
url = {http://eudml.org/doc/77979},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Cohen, Serge
AU - Marty, Renaud
TI - Invariance principle, multifractional gaussian processes and long-range dependence
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 3
SP - 475
EP - 489
AB - 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.
LA - eng
KW - invariance principle; long range dependence; multifractional process; gaussian processes; centered Gaussian field; finite-dimensional convergence; multifractional Gaussian process; long-range dependence
UR - http://eudml.org/doc/77979
ER -

References

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  1. [1] R. J. Adler. The Geometry of Random Fields. Wiley, London, 1981. Zbl0478.60059MR611857
  2. [2] 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. 
  3. [3] A. Ayache and J. Lévy-Véhel. The generalized multifractional Brownian motion. Stat. Inference for Stoch. Process. 3 (2000) 7–18. Zbl0979.60023MR1819282
  4. [4] A. Benassi, S. Cohen and J. Istas. Identifying the multifractional function of a Gaussian process. Statist. Probab. Lett. 39 (1998) 337–345. Zbl0931.60022MR1646220
  5. [5] A. Benassi, S. Cohen and J. Istas. Identification and properties of real harmonizable Lévy motions. Bernoulli 8 (2002) 97–115. Zbl1005.60052MR1884160
  6. [6] A. Benassi, S. Jaffard and D. Roux. Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 (1997) 19–90. Zbl0880.60053MR1462329
  7. [7] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. Zbl0172.21201MR233396
  8. [8] 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. Zbl0965.60073MR1726364
  9. [9] Y. Davydov. The invariance principle for stationary processes. Theory Probab. Appl. 15 (1970) 487-498. Zbl0219.60030MR283872
  10. [10] M. Dekking, J. Lévy-Véhel, E. Lutton and C. Tricot. Fractals: Theory and Applications in Engineering. Springer, London, 1999. Zbl0936.00006MR1726363
  11. [11] C. Lacaux. Real Harmonizable multifractional Lévy motions. Ann. Inst. H. Poincaré, Probab. Statist. 40 (2004) 259–277. Zbl1041.60038MR2060453
  12. [12] B. Mandelbrot, J. V. Ness. Fractional Brownian motion, fractional noises and applications. SIAM Review 10 (1968) 422–437. Zbl0179.47801MR242239
  13. [13] R. Peltier and J. Lévy-Véhel. Multifractional Brownian motion: definition and preliminary results. INRIA research report, RR-2645, 1995. 
  14. [14] A. Philippe, D. Surgailis and M.-C. Viano. Time-varying fractionally integrated processes with nonstationary long memory, Theory Probab. Appl. (2007). To appear. Zbl1167.60326
  15. [15] G. Samorodnitsky and M. S. Taqqu. Stable non-Gaussian Random Processes. Chapman and Hall, New York, 1994. Zbl0925.60027MR1280932

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