Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T08:27:09.146Z Has data issue: false hasContentIssue false
  • English
  • Français

Incremental moments and Hölder exponents of multifractionalmultistable processes

Published online by Cambridge University Press:  08 February 2013

Ronan Le Guével  and
Jacques Lévy Véhel
Ronan Le Guével
Affiliation:
UniversitéParis VI, Laboratoire de Probabilités et Modèles Aléatoires 4 place Jussieu, 75252 Paris Cedex 05, France. [email protected]
Jacques Lévy Véhel
Affiliation:
Regularity team, INRIA Saclay, Parc Orsay Université 4 rue Jacques Monod, Bat P, 91893 Orsay Cedex, France; [email protected]
Get access

Abstract

Multistable processes, that is, processes which are, at each “time”, tangent to a stableprocess, but where the index of stability varies along the path, have been recentlyintroduced as models for phenomena where the intensity of jumps is non constant. In thiswork, we give further results on (multifractional) multistable processes related to theirlocal structure. We show that, under certain conditions, the incremental moments display ascaling behaviour, and that the pointwise Hölder exponent is, as expected, related to thelocal stability index. We compute the precise value of the almost sure Hölder exponent inthe case of the multistable Lévy motion, which turns out to reveal an interestingphenomenon.

Keywords

Localisable processesmultistable processesmultifractional processespointwise Hölder regularity
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 135 - 178
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ayache, A. and Véhel, J. Lévy, The generalized multifractional Brownian motion. Stat. Inference Stoch. Process. 3 (2000) 7–18. Google Scholar
Benassi, A., Jaffard, S. and Roux, D., Gaussian processes and pseudodifferential elliptic operators. Rev. Mat. Iberoam. 13 (1997) 19–89. Google Scholar
Bentkus, V., Juozulynas, A. and Paulauskas, V., Lévy-LePage series representation of stable vectors : convergence in variation. J. Theoret. Probab. 14 (2001) 949–978. Google Scholar
Falconer, K.J., Tangent fields and the local structure of random fields. J. Theoret. Probab. 15 (2002) 731–750. Google Scholar
Falconer, K.J., The local structure of random processes. J. London Math. Soc. 267 (2003) 657–672. Google Scholar
K.J. Falconer and J. Lévy Véhel, Multifractional, multistable, and other processes with prescribed local form. J. Theoret. Probab. (2008) DOI: 10.1007/s10959-008-0147-9.
K.J. Falconer and L. Lining, Multistable random measures and multistable processes. Preprint (2009).
Falconer, K.J., Le Guével, R., and Lévy Véhel, J., Localisable moving average stable and multistable processes. Stoch. Models (2009) 648–672. Google Scholar
Ferguson, T.S. and Klass, M.J., A representation of independent increment processes without Gaussian components. Ann. Math. Stat. 43 (1972) 1634–1643. Google Scholar
Herbin, E., From -parameter fractional Brownian motions to -parameter multifractional Brownian motion. Rocky Mt. J. Math. 36 (2006) 1249–1284. Google Scholar
Herbin, E. and Lévy Véhel, J., Stochastic 2 micro-local analysis. Stoch. Proc. Appl. 119 (2009) 2277–2311. Google Scholar
Kolmogorov, A.N., Wienersche Spiralen und einige andere interessante Kurven in Hilbertchen Raume. Doklady 26 (1940) 115–118. Google Scholar
R. Le Guével and J. Lévy Véhel, A Ferguson–Klass–LePage series representation of multistable multifractional motions and related processes, preprint (2009). Available at http://arxiv.org/abs/0906.5042.
R. Le Page, Multidimensional infinitely divisible variables and processes. I. Stable caseTech. Rep. 292, Dept. Stat., Stanford Univ. (1980).
Le Page, R., Multidimensional infinitely divisible variables and processes, II Probability in Banach Spaces III. Springer, New York, Lect. Notes Math. 860 (1980) 279–284. Google Scholar
M. Ledoux and M. Talagrand, Probability in Banach spaces. Springer-Verlag (1996).
Mandelbrot, B.B. and Van Ness, J., Fractional Brownian motion, fractional noises and applications. SIAM Review 10 (1968) 422–437. Google Scholar
R.F. Peltier and J. Lévy Véhel, 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.
V. Petrov, Limit Theorems of Probability Theory. Oxford Science Publication (1995).
Rosinski, J., On series representations of infinitely divisible random vectors. Ann. Probab. 18 (1990) 405–430. Google Scholar
G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes. Chapman and Hall (1994).
Stoev, S. and Taqqu, M.S., Stochastic properties of the linear multifractional stable motion. Adv. Appl. Probab. 36 (2004) 1085–1115 Google Scholar
Stoev, S. and Taqqu, M.S., Path properties of the linear multifractional stable motion. Fractals 13 (2005) 157–178. Google Scholar
Von Bahr, B. and Essen, C.G., Inequalities for the th absolute moment of a sum of Random variables, 1 ¡= r ¡= 2. Ann. Math. Stat. 36 (1965) 299–303. Google Scholar