no. 3
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

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.

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.

DOI : 10.1214/07-AIHP127
Classification : 60F17, 60G15
Keywords: 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},
     year = {2008},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {3},
     doi = {10.1214/07-AIHP127},
     mrnumber = {2451054},
     zbl = {1176.60021},
     language = {en},
     url = {https://www.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  - https://www.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 https://www.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

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

[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. | Zbl | MR

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

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

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

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

[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. | Zbl | MR

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

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

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

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

[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. | Zbl | MR

Cité par Sources :