Multistable processes, that is, processes which are, at each “time”, tangent to a stable process, but where the index of stability varies along the path, have been recently introduced as models for phenomena where the intensity of jumps is non constant. In this work, we give further results on (multifractional) multistable processes related to their local structure. We show that, under certain conditions, the incremental moments display a scaling behaviour, and that the pointwise Hölder exponent is, as expected, related to the local stability index. We compute the precise value of the almost sure Hölder exponent in the case of the multistable Lévy motion, which turns out to reveal an interesting phenomenon.
Mots clés : localisable processes, multistable processes, multifractional processes, pointwise Hölder regularity
@article{PS_2013__17__135_0,
author = {Le Gu\'evel, Ronan and V\'ehel, Jacques L\'evy},
title = {Incremental moments and {H\"older} exponents of multifractional multistable processes},
journal = {ESAIM: Probability and Statistics},
pages = {135--178},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2011151},
mrnumber = {3021313},
zbl = {1292.60050},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2011151/}
}
TY - JOUR AU - Le Guével, Ronan AU - Véhel, Jacques Lévy TI - Incremental moments and Hölder exponents of multifractional multistable processes JO - ESAIM: Probability and Statistics PY - 2013 SP - 135 EP - 178 VL - 17 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2011151/ DO - 10.1051/ps/2011151 LA - en ID - PS_2013__17__135_0 ER -
%0 Journal Article %A Le Guével, Ronan %A Véhel, Jacques Lévy %T Incremental moments and Hölder exponents of multifractional multistable processes %J ESAIM: Probability and Statistics %D 2013 %P 135-178 %V 17 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2011151/ %R 10.1051/ps/2011151 %G en %F PS_2013__17__135_0
Le Guével, Ronan; Véhel, Jacques Lévy. Incremental moments and Hölder exponents of multifractional multistable processes. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 135-178. doi : 10.1051/ps/2011151. http://archive.numdam.org/articles/10.1051/ps/2011151/
[1] and , The generalized multifractional Brownian motion. Stat. Inference Stoch. Process. 3 (2000) 7-18. | MR | Zbl
[2] , and , Gaussian processes and pseudodifferential elliptic operators. Rev. Mat. Iberoam. 13 (1997) 19-89. | MR | Zbl
[3] , and , Lévy-LePage series representation of stable vectors : convergence in variation. J. Theoret. Probab. 14 (2001) 949-978. | MR | Zbl
[4] , Tangent fields and the local structure of random fields. J. Theoret. Probab. 15 (2002) 731-750. | MR | Zbl
[5] , The local structure of random processes. J. London Math. Soc. 267 (2003) 657-672. | MR | Zbl
[6] and , Multifractional, multistable, and other processes with prescribed local form. J. Theoret. Probab. (2008) DOI: 10.1007/s10959-008-0147-9. | Zbl
[7] and , Multistable random measures and multistable processes. Preprint (2009).
[8] , , and , Localisable moving average stable and multistable processes. Stoch. Models (2009) 648-672. | Zbl
[9] and , A representation of independent increment processes without Gaussian components. Ann. Math. Stat. 43 (1972) 1634-1643. | MR | Zbl
[10] , From -parameter fractional Brownian motions to -parameter multifractional Brownian motion. Rocky Mt. J. Math. 36 (2006) 1249-1284. | MR | Zbl
[11] and , Stochastic 2 micro-local analysis. Stoch. Proc. Appl. 119 (2009) 2277-2311. | Zbl
[12] , Wienersche Spiralen und einige andere interessante Kurven in Hilbertchen Raume. Doklady 26 (1940) 115-118. | JFM | MR | Zbl
[13] and , A Ferguson-Klass-LePage series representation of multistable multifractional motions and related processes, preprint (2009). Available at http://arxiv.org/abs/0906.5042. | Zbl
[14] , Multidimensional infinitely divisible variables and processes. I. Stable caseTech. Rep. 292, Dept. Stat., Stanford Univ. (1980). | MR
[15] , Multidimensional infinitely divisible variables and processes, II Probability in Banach Spaces III. Springer, New York, Lect. Notes Math. 860 (1980) 279-284. | MR | Zbl
[16] and , Probability in Banach spaces. Springer-Verlag (1996). | MR | Zbl
[17] and , Fractional Brownian motion, fractional noises and applications. SIAM Review 10 (1968) 422-437. | MR | Zbl
[18] and , Multifractional Brownian motion : definition and preliminary results. Rapport de recherche de l'INRIA, No. 2645 (1995). Available at : http://www-rocq1.inria.fr/fractales/index.php?page=publications.
[19] , Limit Theorems of Probability Theory. Oxford Science Publication (1995). | MR | Zbl
[20] , On series representations of infinitely divisible random vectors. Ann. Probab. 18 (1990) 405-430. | MR | Zbl
[21] and , Stable Non-Gaussian Random Processes. Chapman and Hall (1994). | MR | Zbl
[22] and , Stochastic properties of the linear multifractional stable motion. Adv. Appl. Probab. 36 (2004) 1085-1115 | MR | Zbl
[23] and , Path properties of the linear multifractional stable motion. Fractals 13 (2005) 157-178. | MR
[24] and , Inequalities for the th absolute moment of a sum of Random variables, 1 ¡= r ¡= 2. Ann. Math. Stat. 36 (1965) 299-303. | MR | Zbl
Cité par Sources :
