Manifold indexed fractional fields
ESAIM: Probability and Statistics, Volume 16 (2012), pp. 222-276.

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

DOI: 10.1051/ps/2011106
Classification: 60G07, 60G15, 60G18
Keywords: self-similarity, stochastic fields, manifold
@article{PS_2012__16__222_0,
     author = {Istas, Jacques},
     title = {Manifold indexed fractional fields},
     journal = {ESAIM: Probability and Statistics},
     pages = {222--276},
     publisher = {EDP-Sciences},
     volume = {16},
     year = {2012},
     doi = {10.1051/ps/2011106},
     mrnumber = {2956575},
     zbl = {1275.60041},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2011106/}
}
TY  - JOUR
AU  - Istas, Jacques
TI  - Manifold indexed fractional fields
JO  - ESAIM: Probability and Statistics
PY  - 2012
SP  - 222
EP  - 276
VL  - 16
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2011106/
DO  - 10.1051/ps/2011106
LA  - en
ID  - PS_2012__16__222_0
ER  - 
%0 Journal Article
%A Istas, Jacques
%T Manifold indexed fractional fields
%J ESAIM: Probability and Statistics
%D 2012
%P 222-276
%V 16
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2011106/
%R 10.1051/ps/2011106
%G en
%F PS_2012__16__222_0
Istas, Jacques. Manifold indexed fractional fields. ESAIM: Probability and Statistics, Volume 16 (2012), pp. 222-276. doi : 10.1051/ps/2011106. http://archive.numdam.org/articles/10.1051/ps/2011106/

[1] P. Abry, P. Gonçalvès and P. Flandrin, Wavelets, spectrum analysis and 1 / f processes. Lect. Note Stat. 103 (1995) 15-29. | Zbl

[2] A. Ayache and J. Lévy-Vehel, The Multifractional Brownian motion. Stat. Inference Stoch. Process. 1 (2000) 7-18. | MR | Zbl

[3] A. Ayache and J. Lévy-Vehel, On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion. Stoc. Proc. Appl. 111 (2004) 119-156. | MR | Zbl

[4] A. Ayache, P. Bertrand and J. Lévy-Vehel, A central limit theorem for the generalized quadratic variation of the step fractional Brownian. Stat. Inference Stoch. Process. 10 (2007) 1-27. | MR | Zbl

[5] J.-M. Bardet, Testing for the presence of self-similarity of Gaussian time series having stationary increments. J. Time Ser. Anal. 25 (2000) 497-515. | MR | Zbl

[6] J.-M. Bardet and P. Bertrand, Identification of the multiscale fractional Brownian motion with biomechanical applications. J. Time Ser. Anal. 28 (2007) 1-52. | MR | Zbl

[7] B. Bekka, P. De La Harpe and A. Valette, Kazhdan's Property (T). Cambridge University Press (2008). | MR | Zbl

[8] A. Benassi, S. Jaffard and D. Roux, Gaussian processes and Pseudodifferential Elliptic operators. Revista Mathematica Iberoam. 13 (1997) 19-90. | EuDML | MR | Zbl

[9] A. Benassi, S. Cohen and J. Istas, Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett. 39 (1998) 337-345. | MR | Zbl

[10] A. Benassi, S. Cohen, J. Istas and S. Jaffard, Identification of filtered white noises. Stoc. Proc. Appl. 75 (1998) 31-49. | MR | Zbl

[11] A. Benassi, P. Bertrand, S. Cohen and J. Istas, Identification of the Hurst index of a step fractional Brownian motion. Stat. Inference Stoch. Process 3 (2000) 101-111. | MR | Zbl

[12] A. Benassi, S. Cohen and J. Istas, Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8 (2002) 97-115. | MR | Zbl

[13] A. Benassi, S. Cohen and J. Istas, On roughness indices for fractional fields. Bernoulli 10 (2004) 357-373. | MR | Zbl

[14] A. Begyn, Quadratic variations along irregular subdivisions for Gaussian processes. Electron. J. Probab. 10 (2005) 691-717. | EuDML | MR | Zbl

[15] A. Begyn, Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. Bernoulli 13 (2007) 712-753. | MR | Zbl

[16] A. Begyn, Functional limit theorems for generalized quadratic variations of Gaussian processes. Stoc. Proc. Appl. 117 (2007) 1848-1869. | MR | Zbl

[17] C. Berzin and J. Leon, Estimating the Hurst parameter. Stat. Inference Stock. Process. 10 (2007) 49-73. | MR | Zbl

[18] A. Bonami and A. Estrade, Anisotropic analysis of Gaussian models. J. Fourier Anal. Appl. 9 (2004) 215-236. | MR | Zbl

[19] V. Borrelli, F. Cazals and J.-M. Morvan, On the angular defect of triangulations and the pointwise approximation of curvatures, curves and surfaces'02. Comput. Aid. Geom. Des. 20 319-341. | MR | Zbl

[20] J. Bretagnolle, D. Dacunha-Castelle and J.-L. Krivine, Lois stables et espaces Lp. Ann. Inst. Henri Poincaré 2 (1969) 231-259. | EuDML | Numdam | MR | Zbl

[21] A. Brouste, J. Istas and S. Lambert-Lacroix, On fractional Gaussian random fields simulation. J. Stat. Soft. 1 (2007) 1-23.

[22] A. Brouste, J. Istas and S. Lambert-Lacroix, On simulation of fractional Brownian motion indexed by a manifold. J. Stat. Soft. 36 (2010).

[23] N. Chentsov, Lévy's Brownian motion of several parameters and generalized white noise. Theory Probab. Appl. 2 (1957) 265-266.

[24] J.-F. Coeurjolly, Simulation and identification of the fractional Brownian motion : a bibliographical and comparative study. J. Stat. Software 5 (2000) 1-53.

[25] J.-F. Coeurjolly, Estimating the parameters of a fractional Brownian Motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 (2001) 199-227. | MR | Zbl

[26] J.-F. Coeurjolly, Identification of multifractional Brownian motion. Bernoulli 11 (2005) 987-1008. | MR | Zbl

[27] J.-F. Coeurjolly, Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles. Ann. Statist. 36 (2008) 1404-1434. | MR | Zbl

[28] J.-F. Coeurjolly and J. Istas, Cramer-Rao bounds for fractional Brownian motions. Stat. Probab. Lett. 53 (2001) 435-447. | MR | Zbl

[29] 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). | MR | Zbl

[30] S. Cohen and J. Istas, An universal estimator of local self-similarity. Preprint (2006).

[31] S. Cohen and J. Istas, Fractional fields : Modelling and statistical applications (Submitted).

[32] S. Cohen and M. Lifshits, Stationary Gaussian random fields on hyperbolic spaces and Euclidean spheres. To appear in ESAIM : PS. | EuDML | Numdam | MR | Zbl

[33] S. Cohen, X. Guyon, O. Perrin and M. Pontier, Singularity functions for fractional processes : application to the fractional brownian sheet. Ann. Inst. Henri Poincaré 42 (2006) 187-205. | EuDML | Numdam | MR | Zbl

[34] D. Dacunha-Castelle and M. Duflo, Probabilités et Statistiques tome 2. Masson, Paris (1983). | MR | Zbl

[35] R. Dalhaus, Efficient parameter estimation for self-similar processes. Ann. Statist. 17 (1989) 1749-1766. | MR | Zbl

[36] I. Daubechies, Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41 (1988) 909-996. | MR | Zbl

[37] S. Dégerine and S. Lambert-Lacroix, Partial autocorrelation function of a nonstationary time series. J. Multiv. Anal. (2003) 46-59. | MR | Zbl

[38] 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. | MR | Zbl

[39] A. Dress, V. Moulton and W. Terhalle, T-theory : An overview, Eur. J. Comb. 17 (1996) 161-175. | MR | Zbl

[40] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher transcendental functions (Bateman manuscript project). McGraw-Hill 2 (1953) | MR | Zbl

[41] K. Falconer, Tangent fields and the local structure of random fields. J. Theor. Probab. 15 (2002) 731-750. | MR | Zbl

[42] K. Falconer, The local structure of random processes. J. Lond. Math. Soc. 67 (2003) 657-672. | MR | Zbl

[43] J. Faraut, Fonction brownienne sur une variété riemannienne. Séminaire de probabilités de Strasbourg 7 (1973) 61-76. | EuDML | Numdam | MR | Zbl

[44] J. Faraut and H. Harzallah, Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier 24 (1974) 171-217. | EuDML | Numdam | MR | Zbl

[45] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, 2nd edition. Springer-Verlag (1993). | Zbl

[46] R. Gangolli, 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. | EuDML | Numdam | MR | Zbl

[47] X. Guyon and J. Leon, Convergence en loi des H-variations d'un processus gaussien stationnaire. Ann. Inst. Henri Poincaré 25 (1989) 265-282. | EuDML | Numdam | MR | Zbl

[48] S. Helgason, Differential Geometry and Symmetric spaces. Academic Press (1962). | MR | Zbl

[49] E. Herbin and E. Merzbach, A set-indexed fractional Brownian motion. J. Theor. Probab. 19 (2006) 337-364. | MR | Zbl

[50] E. Herbin and E. Merzbach, Stationarity and self-similarity characterization of the set-indexed fractional Brownian motion. J. Theor. Probab. 22 (2009) 1010-1029. | MR | Zbl

[51] J. Istas, Spherical and hyperbolic fractional Brownian motion. Electron. Comm. Probab. 10 (2005) 254-262. | EuDML | MR | Zbl

[52] J. Istas, On fractional stable fields indexed by metric spaces. Electron. Comm. Probab. 11 (2006) 242-251. | EuDML | MR | Zbl

[53] J. Istas, Karhunen-Loève expansion of spherical fractional Brownian motions. Stat. Probab. Lett. 76 (2006) 1578-1583. | MR | Zbl

[54] J. Istas, Quadratic variations of spherical fractional Brownian motions, Stoc. Proc. Appl. 117 (2007) 476-486. | MR | Zbl

[55] J. Istas, Identifying the anisotropical function of a d-dimensional Gaussian self-similar process with stationary increments. Stat. Inf. Stoc. Proc. 10-1 (2007) 97-106. | MR | Zbl

[56] J. Istas and C. Lacaux, On locally self-similar fractional random fields indexed by a manifold. preprint. | MR | Zbl

[57] J. Istas and G. Lang, 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. | MR | Zbl

[58] J. Istas and G. Lang, Quadratic variations and estimation of the Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré 33 (1997) 407-436. | EuDML | Numdam | MR | Zbl

[59] J. Kent and A. Wood, Estimating the fractal dimension of a locally self-similar Gaussian process using increments. J. Roy. Statist. Soc. B 59 (1997) 679-700. | MR | Zbl

[60] A. Koldobsky, Schoenberg's problem on positive definite functions. Algebra Anal. 3 (1991) 78-85. | MR | Zbl

[61] A. Koldobsky and Y. Lonke, A short proof of Schoenberg's conjecture on positive definite functions. Bull. Lond. Math. Soc. (1999) 693-699. | MR | Zbl

[62] A. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertsche Raum (German). C. R. (Dokl.) Acad. Sci. URSS 26 (1940) 115-118. | MR | Zbl

[63] C. Lacaux, Real harmonizable multifractional Lévy motions. Ann. Inst. Henri Poincaré 40 (2004) 259-277. | EuDML | Numdam | MR | Zbl

[64] G. Lang and F. Roueff, Semi-parametric estimation of the Hölder exponent of a stationary Gaussian process with minimax rates. Stat. Inf. Stoc. Proc. 4-3 (2001) 283-306. | MR | Zbl

[65] P. Lévy, Processus stochastiques et mouvement Brownien. Gauthier-Vilars (1965). | Zbl

[66] T. Lindstrom, Fractional Brownian fields as integrals of white noise. Bull. Lond. Math. Soc. 25 (1993) 83-88. | MR | Zbl

[67] M. Maejima, A remark on self-similar processes with stationary increments. Can. J. Stat. 14 (1986) 81-82. | MR | Zbl

[68] B.B. Mandelbrot and J.W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968) 422-437. | MR | Zbl

[69] R. Peltier and J. Lévy-Vehel, Multifractional Brownian motion : definition and preliminary results. Rapport de recherche de l'INRIA 2645 (1996).

[70] P. Petersen, Riemannian Geometry. Graduate Texts in Mathematics, Springer (1998). | MR | Zbl

[71] E. Rafajlowicz, Testing (non-)existence of input-output relationships by estimating fractal dimensions. IEEE Trans. Signal Process. 52 (2004) 3151-3159. | MR

[72] G. Robertson, Crofton formulae and geodesic distance in hyperbolic spaces. J. Lie Theory 8 (1998) 163-172. | EuDML | MR | Zbl

[73] G. Robertson and T. Steger, Negative definite kernels and a dynamical characterization of property T for countable groups. Ergod. Theory Dyn. Syst. 18 (1998) 247-253. | MR | Zbl

[74] W. Rudin, Fourier analysis on groups. Wiley (1962). | MR | Zbl

[75] G. Samorodnitsky, Long memory and self-similar processes. Annales de la Faculté des Sciences Toulouse 15 (2006) 107-123. | EuDML | Numdam | MR | Zbl

[76] G. Samorodnitsky and M. Taqqu, Stable non-Gaussian random processes : stochastic models with infinite variance. Chapman & Hall, New York (1994). | MR | Zbl

[77] I. Schönberg, Metric spaces and positive definite functions. Ann. Math. 39 (1938) 811-841. | JFM | Zbl

[78] R. Seeley, Spherical harmonics. Am. Math. Mon. 73 (1966) 115-121. | MR | Zbl

[79] S. Stoev and M. Taqqu, Stochastic properties of the linear multifractional stable motion. Adv. Appl. Prob. 36 (2004) 1085-1115. | MR | Zbl

[80] G. Szego, Orthogonal Polynomials, 4th edition, in Amer. Math. Soc. Providence, RI (1975). | MR

[81] S. Takenaka, Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123 (1991) 1-12. | MR | Zbl

[82] S. Takenaka, I. Kubo and H. Urakawa, Brownian motion parametrized with metric space of constant curvature. Nagoya Math. J. 82 (1981) 131-140. | MR | Zbl

[83] A. Valette, Les représentations uniformément bornées associées à un arbre réel. Bull. Soc. Math. Belgique 42 (1990) 747-760. | MR | Zbl

[84] H. Wang, Two-point homogeneous spaces. Ann. Math. 2 (1952) 177-191. | MR | Zbl

[85] A. Yaglom, Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Probab. Appl. 2 (1957) 273-320.

[86] A. Zaanen, Linear Anal. North Holland Publishing Co (1960).

Cited by Sources: