# Lacunary Fractional Brownian Motion

ESAIM: Probability and Statistics (2012)

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

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topClausel, Marianne. "Lacunary Fractional Brownian Motion." ESAIM: Probability and Statistics 16 (2012): 352-374. <http://eudml.org/doc/222471>.

@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},

month = {8},

pages = {352-374},

publisher = {EDP Sciences},

title = {Lacunary Fractional Brownian Motion},

url = {http://eudml.org/doc/222471},

volume = {16},

year = {2012},

}

TY - JOUR

AU - Clausel, Marianne

TI - Lacunary Fractional Brownian Motion

JO - ESAIM: Probability and Statistics

DA - 2012/8//

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/222471

ER -

## References

top- 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.
- J.M. Bardet and P. Bertrand, Definition, properties and wavelets analysis of Multiscale Fractional Brownian Motion. Fractals15 (2007) 73–87.
- 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).
- A. Benassi, S. Jaffard and D. Roux, Elliptic Gaussian random processes. Revista Matematica Iberoamericana13 (1997) 19–90.
- 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).
- 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.
- 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.
- R.B. Davies and D.S. Harte, Tests for Hurst effect. Biometrika74 (1987) 95–101.
- 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.
- K. Falconer, Fractal Geometry. John Wiley and Sons (1990).
- K. Falconer, Tangent Fields and the local structure of random fields. J. Theor. Prob.15 (2002) 731–750.
- K. Falconer, The local structure of random processes. J. London Math. Soc.67 (2003) 657–672.
- U. Frisch, Turbulence, the legacy of A.N. Kolmogorov. Cambridge University Press (1995).
- J.P. Kahane, Geza Freud and lacunary Fourier series. J. Approx. Theory46 (1986) 51–57.
- I. Karatzas and S.E. Shreve, Brownian Motion and stochastic calculus. Springer-Verlag (1988).
- A.N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C. R. Acad. Sci. URSS26 (1940) 115–118.
- 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).
- 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.
- B.M. Mandelbrot and J. Van Ness, Fractional Brownian Motion, fractional noises and applications. SIAM Rev.10 (1968) 422–437.
- W. Willinger, M.S. Taqqu and V. Teverosky, Stock market price and long-range dependence. Finance and Stochastics1 (1999) 1–14.
- A.T.A. Wood and G. Chan, Simulation of stationary Gaussian processes in [ 0;1 ] d. J. Comput. Graph. Stat.3 (1994) 409–432.

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