Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T14:11:59.252Z Has data issue: false hasContentIssue false
  • English
  • Français

Lacunary Fractional Brownian Motion

Published online by Cambridge University Press:  01 August 2012

Marianne Clausel
Marianne Clausel*
Affiliation:
Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050 du CNRS, Université Paris Est, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France. [email protected]
Get access

Abstract

In this paper, a new class of Gaussian field is introduced called Lacunary Fractional Brownian Motion. Surprisingly we show that usually their tangent fields are not unique at every point. We also investigate the smoothness of the sample paths of Lacunary Fractional Brownian Motion using wavelet analysis.

Keywords

Lacunary Gaussian fieldsnon uniqueness of the tangent fielduniform irregularitywavelets
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 352 - 374
Copyright
© EDP Sciences, SMAI, 2012

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

A. Ayache and J. Lévy-Véhel, Generalized Multifractional Brownian Motion : definition and preliminary results, in Fractals Theory and applications in engineering, edited by M. Dekking, J. Lévy-Véhel, E. Lutton and C. Tricot. Springer (1999) 17–32.
Bardet, J.M. and Bertrand, P., Definition, properties and wavelets analysis of Multiscale Fractional Brownian Motion. Fractals 15 (2007) 73–87. Google Scholar
J.M. Bardet, G. Lang, G. Oppenheim, A. Phillipe, S. Stoev and M.S. Taqqu, Generators of long-range dependent processes : A survey, in Theory and Applications of Long Range Dependance, edited by P. Doukhan M. Oppenheim and G. Taqqu. Birkäuser (2003) 579–623.
M. Basseville and I. Nikiforov, Detection of abrupt changes–Theory and applications. Prentice-Hall (1993).
A. Benassi and S. Deguy, Multi-scale Fractional Motion : definition and identification, Preprint LAIC (1999).
Benassi, A., Jaffard, S. and Roux, D., Elliptic Gaussian random processes. Revista Matematica Iberoamericana 13 (1997) 19–90. Google Scholar
J. Beran, Statistics for Long-Memory processes. Chapman and Hall, London, UK (1994).
Z. Ciesielski, G. Kerkyacharian and B. Roynette, Quelques espaces fonctionnels associés à des processus Gaussiens. Stud. Math. 107 (1993).
M. Clausel, More about uniform irregularity : the wavelet point of view. Preprint (2008).
Collins, J.J. and De Luca, C.J., Open loop and closed loop control of posture : a random walk analysis of center of pressure trajectories, Exp. Brain Res. 9 (1993) 308–318. Google Scholar
H. Csörgö and L. Horvath, Non parametric method for change point problems in Handbook of statistics, edited by P.R. Krishnaiah and C.R. Rao. Elsevier, New York 7 (1988) 403–425.
Davies, R.B. and Harte, D.S., Tests for Hurst effect. Biometrika 74 (1987) 95–101. Google Scholar
Dietrich, C.R. and Newsam, G.N., Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput. 18 (1997) 1088–1107. Google Scholar
K. Falconer, Fractal Geometry. John Wiley and Sons (1990).
Falconer, K., Tangent Fields and the local structure of random fields. J. Theor. Prob. 15 (2002) 731–750. Google Scholar
Falconer, K., The local structure of random processes. J. London Math. Soc. 67 (2003) 657–672. Google Scholar
U. Frisch, Turbulence, the legacy of A.N. Kolmogorov. Cambridge University Press (1995).
Kahane, J.P., Geza Freud and lacunary Fourier series. J. Approx. Theory 46 (1986) 51–57. Google Scholar
I. Karatzas and S.E. Shreve, Brownian Motion and stochastic calculus. Springer-Verlag (1988).
Kolmogorov, A.N., Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C. R. Acad. Sci. URSS 26 (1940) 115–118. Google Scholar
J. Lévy-Vehel and R.F. Peltier, Multifractional Brownian Motion : definition and preliminary results, Rapport de recherche de l’INRIA n° 2645 (1995).
S. Mallat, A wavelet tour of signal processing. Academic Press (1998).
Y. Meyer, Ondelettes et opérateurs. Hermann (1990).
Meyer, Y., Sellan, F. and Taqqu, M.S., Wavelets, generalized white noise and fractional integration : the synthesis of Fractional Brownian Motion. J. Fourier Anal. Appl. 5 (1999) 465–494. Google Scholar
Mandelbrot, B.M. and Van Ness, J., Fractional Brownian Motion, fractional noises and applications. SIAM Rev. 10 (1968) 422–437. Google Scholar
Willinger, W., Taqqu, M.S. and Teverosky, V., Stock market price and long-range dependence. Finance and Stochastics 1 (1999) 1–14. Google Scholar
Wood, A.T.A. and Chan, G., Simulation of stationary Gaussian processes in  [ 0;1 ] d. J. Comput. Graph. Stat. 3 (1994) 409–432. Google Scholar