Lacunary Fractional brownian Motion

Marianne Clausel

ESAIM: Probability and Statistics (2012)

  • Volume: 16, page 352-374
  • ISSN: 1292-8100

Abstract

top
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.

How to cite

top

Clausel, Marianne. "Lacunary Fractional brownian Motion." ESAIM: Probability and Statistics 16 (2012): 352-374. <http://eudml.org/doc/274375>.

@article{Clausel2012,
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.},
author = {Clausel, Marianne},
journal = {ESAIM: Probability and Statistics},
keywords = {lacunary gaussian fields; non uniqueness of the tangent field; uniform irregularity; wavelets; lacunary Gaussian fields},
language = {eng},
pages = {352-374},
publisher = {EDP-Sciences},
title = {Lacunary Fractional brownian Motion},
url = {http://eudml.org/doc/274375},
volume = {16},
year = {2012},
}

TY - JOUR
AU - Clausel, Marianne
TI - Lacunary Fractional brownian Motion
JO - ESAIM: Probability and Statistics
PY - 2012
PB - EDP-Sciences
VL - 16
SP - 352
EP - 374
AB - 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.
LA - eng
KW - lacunary gaussian fields; non uniqueness of the tangent field; uniform irregularity; wavelets; lacunary Gaussian fields
UR - http://eudml.org/doc/274375
ER -

References

top
  1. [1] 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. Zbl0964.60046MR1726365
  2. [2] J.M. Bardet and P. Bertrand, Definition, properties and wavelets analysis of Multiscale Fractional Brownian Motion. Fractals15 (2007) 73–87. Zbl1142.60329MR2281946
  3. [3] 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. Zbl1031.65010MR1957509
  4. [4] M. Basseville and I. Nikiforov, Detection of abrupt changes–Theory and applications. Prentice-Hall (1993). MR1210954
  5. [5] A. Benassi and S. Deguy, Multi-scale Fractional Motion : definition and identification, Preprint LAIC (1999). 
  6. [6] A. Benassi, S. Jaffard and D. Roux, Elliptic Gaussian random processes. Revista Matematica Iberoamericana13 (1997) 19–90. Zbl0880.60053MR1462329
  7. [7] J. Beran, Statistics for Long-Memory processes. Chapman and Hall, London, UK (1994). Zbl0869.60045MR1304490
  8. [8] Z. Ciesielski, G. Kerkyacharian and B. Roynette, Quelques espaces fonctionnels associés à des processus Gaussiens. Stud. Math. 107 (1993). Zbl0809.60004MR1244574
  9. [9] M. Clausel, More about uniform irregularity : the wavelet point of view. Preprint (2008). 
  10. [10] J.J. Collins and C.J. De Luca, Open loop and closed loop control of posture : a random walk analysis of center of pressure trajectories, Exp. Brain Res.9 (1993) 308–318. 
  11. [11] 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. 
  12. [12] R.B. Davies and D.S. Harte, Tests for Hurst effect. Biometrika74 (1987) 95–101. Zbl0612.62123MR885922
  13. [13] C.R. Dietrich and G.N. Newsam, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput.18 (1997) 1088–1107. Zbl0890.65149MR1453559
  14. [14] K. Falconer, Fractal Geometry. John Wiley and Sons (1990). Zbl0689.28003MR1102677
  15. [15] K. Falconer, Tangent Fields and the local structure of random fields. J. Theor. Prob.15 (2002) 731–750. Zbl1013.60028MR1922445
  16. [16] K. Falconer, The local structure of random processes. J. London Math. Soc.67 (2003) 657–672. Zbl1054.28003MR1967698
  17. [17] U. Frisch, Turbulence, the legacy of A.N. Kolmogorov. Cambridge University Press (1995). Zbl0832.76001MR1428905
  18. [18] J.P. Kahane, Geza Freud and lacunary Fourier series. J. Approx. Theory 46 (1986) 51–57. Zbl0621.42013MR835726
  19. [19] I. Karatzas and S.E. Shreve, Brownian Motion and stochastic calculus. Springer-Verlag (1988). Zbl0734.60060MR917065
  20. [20] A.N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C. R. Acad. Sci. URSS26 (1940) 115–118. Zbl66.0552.03MR3441JFM66.0552.03
  21. [21] J. Lévy-Vehel and R.F. Peltier, Multifractional Brownian Motion : definition and preliminary results, Rapport de recherche de l’INRIA n° 2645 (1995). 
  22. [22] S. Mallat, A wavelet tour of signal processing. Academic Press (1998). Zbl0998.94510MR1614527
  23. [23] Y. Meyer, Ondelettes et opérateurs. Hermann (1990). Zbl0694.41037MR1085487
  24. [24] Y. Meyer, F. Sellan and M.S. Taqqu, Wavelets, generalized white noise and fractional integration : the synthesis of Fractional Brownian Motion. J. Fourier Anal. Appl.5 (1999) 465–494. Zbl0948.60026MR1755100
  25. [25] B.M. Mandelbrot and J. Van Ness, Fractional Brownian Motion, fractional noises and applications. SIAM Rev.10 (1968) 422–437. Zbl0179.47801MR242239
  26. [26] W. Willinger, M.S. Taqqu and V. Teverosky, Stock market price and long-range dependence. Finance and Stochastics1 (1999) 1–14. Zbl0924.90029
  27. [27] A.T.A. Wood and G. Chan, Simulation of stationary Gaussian processes in [ 0;1 ] d. J. Comput. Graph. Stat.3 (1994) 409–432. MR1323050

NotesEmbed ?

top

You must be logged in to post comments.