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Manifold indexed fractional fields∗

Published online by Cambridge University Press:  11 July 2012

Jacques Istas
Jacques Istas*
Affiliation:
Laboratoire Jean Kuntzmann, Université de Grenoble et CNRS, 38041 Grenoble Cedex 9, France. [email protected]
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Abstract

(Local) self-similarity is a seminal concept, especially for Euclidean random fields. Westudy in this paper the extension of these notions to manifold indexed fields. We giveconditions on the (local) self-similarity index that ensure the existence of fractionalfields. Moreover, we explain how to identify the self-similar index. We describe a way ofsimulating Gaussian fractional fields.

Keywords

Self-similaritystochastic fieldsmanifold
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 222 - 276
Copyright
© EDP Sciences, SMAI, 2012

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References

Abry, P., Gonçalvès, P. and Flandrin, P., Wavelets, spectrum analysis and 1 / f processes. Lect. Note Stat. 103 (1995) 15–29. Google Scholar
Ayache, A. and Lévy-Vehel, J., The Multifractional Brownian motion. Stat. Inference Stoch. Process. 1 (2000) 7–18. Google Scholar
Ayache, A. and Lévy-Vehel, J., On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion. Stoc. Proc. Appl. 111 (2004) 119–156. Google Scholar
Ayache, A., Bertrand, P. and Lévy-Vehel, J., A central limit theorem for the generalized quadratic variation of the step fractional Brownian. Stat. Inference Stoch. Process. 10 (2007) 1–27. Google Scholar
Bardet, J.-M., Testing for the presence of self-similarity of Gaussian time series having stationary increments. J. Time Ser. Anal. 25 (2000) 497–515. Google Scholar
Bardet, J.-M. and Bertrand, P., Identification of the multiscale fractional Brownian motion with biomechanical applications. J. Time Ser. Anal. 28 (2007) 1–52. Google Scholar
B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s Property (T). Cambridge University Press (2008).
Benassi, A., Jaffard, S. and Roux, D., Gaussian processes and Pseudodifferential Elliptic operators. Revista Mathematica Iberoam. 13 (1997) 19–90. Google Scholar
Benassi, A., Cohen, S. and Istas, J., Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett. 39 (1998) 337–345. Google Scholar
Benassi, A., Cohen, S., Istas, J. and Jaffard, S., Identification of filtered white noises. Stoc. Proc. Appl. 75 (1998) 31–49. Google Scholar
Benassi, A., Bertrand, P., Cohen, S. and Istas, J., Identification of the Hurst index of a step fractional Brownian motion. Stat. Inference Stoch. Process 3 (2000) 101–111. Google Scholar
Benassi, A., Cohen, S. and Istas, J., Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8 (2002) 97–115. Google Scholar
Benassi, A., Cohen, S. and Istas, J., On roughness indices for fractional fields. Bernoulli 10 (2004) 357–373. Google Scholar
Begyn, A., Quadratic variations along irregular subdivisions for Gaussian processes. Electron. J. Probab. 10 (2005) 691–717. Google Scholar
Begyn, A., Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. Bernoulli 13 (2007) 712–753. Google Scholar
Begyn, A., Functional limit theorems for generalized quadratic variations of Gaussian processes. Stoc. Proc. Appl. 117 (2007) 1848–1869. Google Scholar
Berzin, C. and Leon, J., Estimating the Hurst parameter. Stat. Inference Stock. Process. 10 (2007) 49–73. Google Scholar
Bonami, A. and Estrade, A., Anisotropic analysis of Gaussian models. J. Fourier Anal. Appl. 9 (2004) 215–236. Google Scholar
Borrelli, V., Cazals, F. and Morvan, J.-M., On the angular defect of triangulations and the pointwise approximation of curvatures, curves and surfaces’02. Comput. Aid. Geom. Des. 20 319–341.
Bretagnolle, J., Dacunha-Castelle, D. and Krivine, J.-L., Lois stables et espaces L p. Ann. Inst. Henri Poincaré 2 (1969) 231–259. Google Scholar
Brouste, A., Istas, J. and Lambert-Lacroix, S., On fractional Gaussian random fields simulation. J. Stat. Soft. 1 (2007) 1–23. Google Scholar
A. Brouste, J. Istas and S. Lambert-Lacroix, On simulation of fractional Brownian motion indexed by a manifold. J. Stat. Soft. 36 (2010).
Chentsov, N., Lévy’s Brownian motion of several parameters and generalized white noise. Theory Probab. Appl. 2 (1957) 265–266. Google Scholar
Coeurjolly, J.-F., Simulation and identification of the fractional Brownian motion : a bibliographical and comparative study. J. Stat. Software 5 (2000) 1–53. Google Scholar
Coeurjolly, J.-F., Estimating the parameters of a fractional Brownian Motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 (2001) 199–227. Google Scholar
Coeurjolly, J.-F., Identification of multifractional Brownian motion. Bernoulli 11 (2005) 987–1008. Google Scholar
Coeurjolly, J.-F., Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles. Ann. Statist. 36 (2008) 1404–1434. Google Scholar
Coeurjolly, J.-F. and Istas, J., Cramer-Rao bounds for fractional Brownian motions. Stat. Probab. Lett. 53 (2001) 435–447. Google Scholar
S. Cohen, From self-similarity to local self-similarity : the estimation problem. Fractal in Engineering, edited by J. Lévy-Vehel and C. Tricot. Springer Verlag, Delft (1999).
S. Cohen and J. Istas, An universal estimator of local self-similarity. Preprint (2006).
S. Cohen and J. Istas, Fractional fields : Modelling and statistical applications (Submitted).
S. Cohen and M. Lifshits, Stationary Gaussian random fields on hyperbolic spaces and Euclidean spheres. To appear in ESAIM : PS.
Cohen, S., Guyon, X., Perrin, O. and Pontier, M., Singularity functions for fractional processes : application to the fractional brownian sheet. Ann. Inst. Henri Poincaré 42 (2006) 187–205. Google Scholar
D. Dacunha-Castelle and M. Duflo, Probabilités et Statistiques tome 2. Masson, Paris (1983).
Dalhaus, R., Efficient parameter estimation for self-similar processes. Ann. Statist. 17 (1989) 1749–1766. Google Scholar
Daubechies, I., Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41 (1988) 909–996. Google Scholar
Dégerine, S. and Lambert-Lacroix, S., Partial autocorrelation function of a nonstationary time series. J. Multiv. Anal. (2003) 46–59. Google Scholar
R.L. Dobrushin, Automodel generalized random fields and their renorm group, in Multicomponent Random Systems, edited by R.L. Dobrushin and Ya. G. Sinai. Dekker, New York (1980) 153–198.
Dress, A., Moulton, V. and Terhalle, W., T-theory : An overview, Eur. J. Comb. 17 (1996) 161–175. Google Scholar
A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher transcendental functions (Bateman manuscript project). McGraw-Hill 2 (1953)
Falconer, K., Tangent fields and the local structure of random fields. J. Theor. Probab. 15 (2002) 731–750. Google Scholar
Falconer, K., The local structure of random processes. J. Lond. Math. Soc. 67 (2003) 657–672. Google Scholar
Faraut, J., Fonction brownienne sur une variété riemannienne. Séminaire de probabilités de Strasbourg 7 (1973) 61–76. Google Scholar
Faraut, J. and Harzallah, H., Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier 24 (1974) 171–217. Google Scholar
S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, 2nd edition. Springer-Verlag (1993).
Gangolli, R., Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters. Ann. Inst. Henri Poincaré 3 (1967) 121–226. Google Scholar
Guyon, X. and Leon, J., Convergence en loi des H-variations d’un processus gaussien stationnaire. Ann. Inst. Henri Poincaré 25 (1989) 265–282. Google Scholar
S. Helgason, Differential Geometry and Symmetric spaces. Academic Press (1962).
Herbin, E. and Merzbach, E., A set-indexed fractional Brownian motion. J. Theor. Probab. 19 (2006) 337–364. Google Scholar
Herbin, E. and Merzbach, E., Stationarity and self-similarity characterization of the set-indexed fractional Brownian motion. J. Theor. Probab. 22 (2009) 1010–1029. Google Scholar
Istas, J., Spherical and hyperbolic fractional Brownian motion. Electron. Comm. Probab. 10 (2005) 254–262. Google Scholar
Istas, J., On fractional stable fields indexed by metric spaces. Electron. Comm. Probab. 11 (2006) 242–251. Google Scholar
Istas, J., Karhunen-Loève expansion of spherical fractional Brownian motions. Stat. Probab. Lett. 76 (2006) 1578–1583. Google Scholar
Istas, J., Quadratic variations of spherical fractional Brownian motions, Stoc. Proc. Appl. 117 (2007) 476–486. Google Scholar
Istas, J., Identifying the anisotropical function of a d-dimensional Gaussian self-similar process with stationary increments. Stat. Inf. Stoc. Proc. 10-1 (2007) 97–106. Google Scholar
J. Istas and C. Lacaux, On locally self-similar fractional random fields indexed by a manifold. preprint.
Istas, J. and Lang, G., Variations quadratiques et estimation de l’exposant de Hölder local d’un processus gaussien. C. R. Acad. Sci. Sér. I Paris 319 (1994) 201–206. Google Scholar
Istas, J. and Lang, G., Quadratic variations and estimation of the Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré 33 (1997) 407–436. Google Scholar
Kent, J. and Wood, A., Estimating the fractal dimension of a locally self-similar Gaussian process using increments. J. Roy. Statist. Soc. B 59 (1997) 679–700. Google Scholar
Koldobsky, A., Schoenberg’s problem on positive definite functions. Algebra Anal. 3 (1991) 78–85. Google Scholar
Koldobsky, A. and Lonke, Y., A short proof of Schoenberg’s conjecture on positive definite functions. Bull. Lond. Math. Soc. (1999) 693–699. Google Scholar
Kolmogorov, A., Wienersche Spiralen und einige andere interessante Kurven im Hilbertsche Raum (German). C. R. (Dokl.) Acad. Sci. URSS 26 (1940) 115–118. Google Scholar
Lacaux, C., Real harmonizable multifractional Lévy motions. Ann. Inst. Henri Poincaré 40 (2004) 259–277. Google Scholar
Lang, G. and Roueff, F., Semi-parametric estimation of the Hölder exponent of a stationary Gaussian process with minimax rates. Stat. Inf. Stoc. Proc. 4-3 (2001) 283–306. Google Scholar
P. Lévy, Processus stochastiques et mouvement Brownien. Gauthier-Vilars (1965).
Lindstrom, T., Fractional Brownian fields as integrals of white noise. Bull. Lond. Math. Soc. 25 (1993) 83–88. Google Scholar
Maejima, M., A remark on self-similar processes with stationary increments. Can. J. Stat. 14 (1986) 81–82. Google Scholar
Mandelbrot, B.B. and Van Ness, J.W., Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968) 422–437. Google Scholar
R. Peltier and J. Lévy-Vehel, Multifractional Brownian motion : definition and preliminary results. Rapport de recherche de l’INRIA 2645 (1996).
P. Petersen, Riemannian Geometry. Graduate Texts in Mathematics, Springer (1998).
Rafajlowicz, E., Testing (non-)existence of input-output relationships by estimating fractal dimensions. IEEE Trans. Signal Process. 52 (2004) 3151–3159. Google Scholar
Robertson, G., Crofton formulae and geodesic distance in hyperbolic spaces. J. Lie Theory 8 (1998) 163–172. Google Scholar
Robertson, G. and Steger, T., Negative definite kernels and a dynamical characterization of property T for countable groups. Ergod. Theory Dyn. Syst. 18 (1998) 247–253. Google Scholar
W. Rudin, Fourier analysis on groups. Wiley (1962).
Samorodnitsky, G., Long memory and self-similar processes. Annales de la Faculté des Sciences Toulouse 15 (2006) 107–123. Google Scholar
G. Samorodnitsky and M. Taqqu, Stable non-Gaussian random processes : stochastic models with infinite variance. Chapman & Hall, New York (1994).
Schönberg, I., Metric spaces and positive definite functions. Ann. Math. 39 (1938) 811–841. Google Scholar
Seeley, R., Spherical harmonics. Am. Math. Mon. 73 (1966) 115–121. Google Scholar
Stoev, S. and Taqqu, M., Stochastic properties of the linear multifractional stable motion. Adv. Appl. Prob. 36 (2004) 1085–1115. Google Scholar
G. Szego, Orthogonal Polynomials, 4th edition, in Amer. Math. Soc. Providence, RI (1975).
Takenaka, S., Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123 (1991) 1–12. Google Scholar
Takenaka, S., Kubo, I. and Urakawa, H., Brownian motion parametrized with metric space of constant curvature. Nagoya Math. J. 82 (1981) 131–140. Google Scholar
Valette, A., Les représentations uniformément bornées associées à un arbre réel. Bull. Soc. Math. Belgique 42 (1990) 747–760. Google Scholar
Wang, H., Two-point homogeneous spaces. Ann. Math. 2 (1952) 177–191. Google Scholar
Yaglom, A., Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Probab. Appl. 2 (1957) 273–320. Google Scholar
A. Zaanen, Linear Anal. North Holland Publishing Co (1960).