Invariance principle, multifractional gaussian processes and long-range dependence
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 3, pp. 475-489.

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.

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.

DOI : 10.1214/07-AIHP127
Classification : 60F17, 60G15
Mots-clés : invariance principle, long range dependence, multifractional process, gaussian processes 
@article{AIHPB_2008__44_3_475_0,
     author = {Cohen, Serge and Marty, Renaud},
     title = {Invariance principle, multifractional gaussian processes and long-range dependence},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {475--489},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {3},
     year = {2008},
     doi = {10.1214/07-AIHP127},
     mrnumber = {2451054},
     zbl = {1176.60021},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/07-AIHP127/}
}
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
SP  - 475
EP  - 489
VL  - 44
IS  - 3
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/07-AIHP127/
DO  - 10.1214/07-AIHP127
LA  - en
ID  - AIHPB_2008__44_3_475_0
ER  - 
%0 Journal Article
%A Cohen, Serge
%A Marty, Renaud
%T Invariance principle, multifractional gaussian processes and long-range dependence
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2008
%P 475-489
%V 44
%N 3
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/07-AIHP127/
%R 10.1214/07-AIHP127
%G en
%F AIHPB_2008__44_3_475_0
Cohen, Serge; Marty, Renaud. Invariance principle, multifractional gaussian processes and long-range dependence. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 3, pp. 475-489. doi : 10.1214/07-AIHP127. http://archive.numdam.org/articles/10.1214/07-AIHP127/

[1] R. J. Adler. The Geometry of Random Fields. Wiley, London, 1981. | MR | Zbl

[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] A. Ayache and J. Lévy-Véhel. The generalized multifractional Brownian motion. Stat. Inference for Stoch. Process. 3 (2000) 7-18. | MR | Zbl

[4] A. Benassi, S. Cohen and J. Istas. Identifying the multifractional function of a Gaussian process. Statist. Probab. Lett. 39 (1998) 337-345. | MR | Zbl

[5] A. Benassi, S. Cohen and J. Istas. Identification and properties of real harmonizable Lévy motions. Bernoulli 8 (2002) 97-115. | MR | Zbl

[6] A. Benassi, S. Jaffard and D. Roux. Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 (1997) 19-90. | EuDML | MR | Zbl

[7] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. | MR | Zbl

[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. | MR | Zbl

[9] Y. Davydov. The invariance principle for stationary processes. Theory Probab. Appl. 15 (1970) 487-498. | MR | Zbl

[10] M. Dekking, J. Lévy-Véhel, E. Lutton and C. Tricot. Fractals: Theory and Applications in Engineering. Springer, London, 1999. | MR | Zbl

[11] C. Lacaux. Real Harmonizable multifractional Lévy motions. Ann. Inst. H. Poincaré, Probab. Statist. 40 (2004) 259-277. | Numdam | MR | Zbl

[12] B. Mandelbrot, J. V. Ness. Fractional Brownian motion, fractional noises and applications. SIAM Review 10 (1968) 422-437. | MR | Zbl

[13] R. Peltier and J. Lévy-Véhel. Multifractional Brownian motion: definition and preliminary results. INRIA research report, RR-2645, 1995.

[14] A. Philippe, D. Surgailis and M.-C. Viano. Time-varying fractionally integrated processes with nonstationary long memory, Theory Probab. Appl. (2007). To appear. | Zbl

[15] G. Samorodnitsky and M. S. Taqqu. Stable non-Gaussian Random Processes. Chapman and Hall, New York, 1994. | MR | Zbl

Cité par Sources :