Manifold indexed fractional fields
ESAIM: Probability and Statistics (2012)
- Volume: 16, page 222-276
- ISSN: 1292-8100
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topIstas, Jacques. "Manifold indexed fractional fields." ESAIM: Probability and Statistics 16 (2012): 222-276. <http://eudml.org/doc/274362>.
@article{Istas2012,
abstract = {(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.},
author = {Istas, Jacques},
journal = {ESAIM: Probability and Statistics},
keywords = {self-similarity; stochastic fields; manifold; Gaussian field; stable field; Euclidean random field},
language = {eng},
pages = {222-276},
publisher = {EDP-Sciences},
title = {Manifold indexed fractional fields},
url = {http://eudml.org/doc/274362},
volume = {16},
year = {2012},
}
TY - JOUR
AU - Istas, Jacques
TI - Manifold indexed fractional fields
JO - ESAIM: Probability and Statistics
PY - 2012
PB - EDP-Sciences
VL - 16
SP - 222
EP - 276
AB - (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.
LA - eng
KW - self-similarity; stochastic fields; manifold; Gaussian field; stable field; Euclidean random field
UR - http://eudml.org/doc/274362
ER -
References
top- [1] P. Abry, P. Gonçalvès and P. Flandrin, Wavelets, spectrum analysis and 1 / f processes. Lect. Note Stat.103 (1995) 15–29. Zbl0828.62083
- [2] A. Ayache and J. Lévy-Vehel, The Multifractional Brownian motion. Stat. Inference Stoch. Process.1 (2000) 7–18. Zbl0979.60023
- [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. Zbl1079.60029MR2049572
- [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. Zbl1115.60024MR2269602
- [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. Zbl0972.62070MR1794484
- [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. Zbl1164.62034MR2332850
- [7] B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s Property (T). Cambridge University Press (2008). Zbl1146.22009
- [8] A. Benassi, S. Jaffard and D. Roux, Gaussian processes and Pseudodifferential Elliptic operators. Revista Mathematica Iberoam.13 (1997) 19–90. Zbl0880.60053MR1462329
- [9] A. Benassi, S. Cohen and J. Istas, Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett.39 (1998) 337–345. Zbl0931.60022MR1646220
- [10] A. Benassi, S. Cohen, J. Istas and S. Jaffard, Identification of filtered white noises. Stoc. Proc. Appl.75 (1998) 31–49. Zbl0932.60037MR1629014
- [11] A. Benassi, P. Bertrand, S. Cohen and J. Istas, Identification of the Hurst index of a step fractional Brownian motion. Stat. Inference Stoch. Process3 (2000) 101–111. Zbl0982.60081MR1819289
- [12] A. Benassi, S. Cohen and J. Istas, Identification and properties of real harmonizable fractional Lévy motions. Bernoulli8 (2002) 97–115. Zbl1005.60052MR1884160
- [13] A. Benassi, S. Cohen and J. Istas, On roughness indices for fractional fields. Bernoulli10 (2004) 357–373. Zbl1062.60052MR2046778
- [14] A. Begyn, Quadratic variations along irregular subdivisions for Gaussian processes. Electron. J. Probab.10 (2005) 691–717. Zbl1109.60024MR2164027
- [15] A. Begyn, Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. Bernoulli13 (2007) 712–753. Zbl1143.60030MR2348748
- [16] A. Begyn, Functional limit theorems for generalized quadratic variations of Gaussian processes. Stoc. Proc. Appl.117 (2007) 1848–1869. Zbl1129.60032MR2437732
- [17] C. Berzin and J. Leon, Estimating the Hurst parameter. Stat. Inference Stock. Process.10 (2007) 49–73. Zbl1110.62110MR2269604
- [18] A. Bonami and A. Estrade, Anisotropic analysis of Gaussian models. J. Fourier Anal. Appl.9 (2004) 215–236. Zbl1034.60038MR1988750
- [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. Zbl1069.65544MR2007708
- [20] J. Bretagnolle, D. Dacunha-Castelle and J.-L. Krivine, Lois stables et espaces Lp. Ann. Inst. Henri Poincaré2 (1969) 231–259. Zbl0139.33501MR203757
- [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. Software5 (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. Zbl0984.62058MR1856174
- [26] J.-F. Coeurjolly, Identification of multifractional Brownian motion. Bernoulli11 (2005) 987–1008. Zbl1098.62109MR2188838
- [27] J.-F. Coeurjolly, Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles. Ann. Statist.36 (2008) 1404–1434. Zbl1157.60034MR2418662
- [28] J.-F. Coeurjolly and J. Istas, Cramer-Rao bounds for fractional Brownian motions. Stat. Probab. Lett.53 (2001) 435–447. Zbl1092.62574MR1856169
- [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). Zbl0965.60073MR1726364
- [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). Zbl1279.60006
- [32] S. Cohen and M. Lifshits, Stationary Gaussian random fields on hyperbolic spaces and Euclidean spheres. To appear in ESAIM : PS. Zbl1275.60038MR2946126
- [33] S. Cohen, X. Guyon, O. Perrin and M. Pontier, Singularity functions for fractional processes : application to the fractional brownian sheet. Ann. Inst. Henri Poincaré42 (2006) 187–205. Zbl1095.60011MR2199797
- [34] D. Dacunha-Castelle and M. Duflo, Probabilités et Statistiques tome 2. Masson, Paris (1983). Zbl0535.62004MR732786
- [35] R. Dalhaus, Efficient parameter estimation for self-similar processes. Ann. Statist.17 (1989) 1749–1766. Zbl0703.62091MR1026311
- [36] I. Daubechies, Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math.41 (1988) 909–996. Zbl0644.42026MR951745
- [37] S. Dégerine and S. Lambert-Lacroix, Partial autocorrelation function of a nonstationary time series. J. Multiv. Anal. (2003) 46–59. Zbl1033.60046
- [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. Zbl0499.60047MR599535
- [39] A. Dress, V. Moulton and W. Terhalle, T-theory : An overview, Eur. J. Comb.17 (1996) 161–175. Zbl0853.54027MR1379369
- [40] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher transcendental functions (Bateman manuscript project). McGraw-Hill 2 (1953) Zbl0051.30303MR58756
- [41] K. Falconer, Tangent fields and the local structure of random fields. J. Theor. Probab.15 (2002) 731–750. Zbl1013.60028MR1922445
- [42] K. Falconer, The local structure of random processes. J. Lond. Math. Soc.67 (2003) 657–672. Zbl1054.28003MR1967698
- [43] J. Faraut, Fonction brownienne sur une variété riemannienne. Séminaire de probabilités de Strasbourg7 (1973) 61–76. Zbl0261.60052MR391284
- [44] J. Faraut and H. Harzallah, Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier24 (1974) 171–217. Zbl0265.43013MR365042
- [45] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, 2nd edition. Springer-Verlag (1993). Zbl0636.53001
- [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 Poincaré3 (1967) 121–226. Zbl0157.24902MR215331
- [47] X. Guyon and J. Leon, Convergence en loi des H-variations d’un processus gaussien stationnaire. Ann. Inst. Henri Poincaré25 (1989) 265–282. Zbl0691.60017MR1023952
- [48] S. Helgason, Differential Geometry and Symmetric spaces. Academic Press (1962). Zbl0111.18101MR145455
- [49] E. Herbin and E. Merzbach, A set-indexed fractional Brownian motion. J. Theor. Probab.19 (2006) 337–364. Zbl1120.60035MR2283380
- [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. Zbl1197.60040MR2558663
- [51] J. Istas, Spherical and hyperbolic fractional Brownian motion. Electron. Comm. Probab.10 (2005) 254–262. Zbl1112.60029MR2198600
- [52] J. Istas, On fractional stable fields indexed by metric spaces. Electron. Comm. Probab.11 (2006) 242–251. Zbl1110.60032MR2266715
- [53] J. Istas, Karhunen-Loève expansion of spherical fractional Brownian motions. Stat. Probab. Lett.76 (2006) 1578–1583. Zbl1096.60019MR2245581
- [54] J. Istas, Quadratic variations of spherical fractional Brownian motions, Stoc. Proc. Appl.117 (2007) 476–486. Zbl1112.60030MR2305382
- [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. Zbl1119.60027MR2269606
- [56] J. Istas and C. Lacaux, On locally self-similar fractional random fields indexed by a manifold. preprint. Zbl1293.60056MR3176470
- [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 Paris319 (1994) 201–206. Zbl0803.60038MR1288403
- [58] J. Istas and G. Lang, Quadratic variations and estimation of the Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré33 (1997) 407–436. Zbl0882.60032MR1465796
- [59] J. Kent and A. Wood, Estimating the fractal dimension of a locally self-similar Gaussian process using increments. J. Roy. Statist. Soc. B59 (1997) 679–700. Zbl0889.62072MR1452033
- [60] A. Koldobsky, Schoenberg’s problem on positive definite functions. Algebra Anal.3 (1991) 78–85. Zbl0741.60010MR1150554
- [61] A. Koldobsky and Y. Lonke, A short proof of Schoenberg’s conjecture on positive definite functions. Bull. Lond. Math. Soc. (1999) 693–699. Zbl1020.42005MR1711028
- [62] A. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertsche Raum (German). C. R. (Dokl.) Acad. Sci. URSS 26 (1940) 115–118. Zbl0022.36001MR3441
- [63] C. Lacaux, Real harmonizable multifractional Lévy motions. Ann. Inst. Henri Poincaré40 (2004) 259–277. Zbl1041.60038MR2060453
- [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. Zbl1008.62081MR1868724
- [65] P. Lévy, Processus stochastiques et mouvement Brownien. Gauthier-Vilars (1965). Zbl0034.22603
- [66] T. Lindstrom, Fractional Brownian fields as integrals of white noise. Bull. Lond. Math. Soc.25 (1993) 83–88. Zbl0741.60031MR1190370
- [67] M. Maejima, A remark on self-similar processes with stationary increments. Can. J. Stat.14 (1986) 81–82. Zbl0589.60044MR839294
- [68] B.B. Mandelbrot and J.W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev.10 (1968) 422–437. Zbl0179.47801MR242239
- [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). Zbl1220.53002MR1480173
- [71] E. Rafajlowicz, Testing (non-)existence of input-output relationships by estimating fractal dimensions. IEEE Trans. Signal Process.52 (2004) 3151–3159. MR2095596
- [72] G. Robertson, Crofton formulae and geodesic distance in hyperbolic spaces. J. Lie Theory8 (1998) 163–172. Zbl0891.43005MR1616751
- [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. Zbl0966.22005MR1609459
- [74] W. Rudin, Fourier analysis on groups. Wiley (1962). Zbl0698.43001MR152834
- [75] G. Samorodnitsky, Long memory and self-similar processes. Annales de la Faculté des Sciences Toulouse15 (2006) 107–123. Zbl1255.60060MR2225749
- [76] G. Samorodnitsky and M. Taqqu, Stable non-Gaussian random processes : stochastic models with infinite variance. Chapman & Hall, New York (1994). Zbl0925.60027MR1280932
- [77] I. Schönberg, Metric spaces and positive definite functions. Ann. Math.39 (1938) 811–841. Zbl0019.41503JFM64.0617.03
- [78] R. Seeley, Spherical harmonics. Am. Math. Mon.73 (1966) 115–121. Zbl0142.03503MR201695
- [79] S. Stoev and M. Taqqu, Stochastic properties of the linear multifractional stable motion. Adv. Appl. Prob.36 (2004) 1085–1115. Zbl1068.60057MR2119856
- [80] G. Szego, Orthogonal Polynomials, 4th edition, in Amer. Math. Soc. Providence, RI (1975). MR372517
- [81] S. Takenaka, Integral-geometric construction of self-similar stable processes. Nagoya Math. J.123 (1991) 1–12. Zbl0757.60035MR1126180
- [82] S. Takenaka, I. Kubo and H. Urakawa, Brownian motion parametrized with metric space of constant curvature. Nagoya Math. J.82 (1981) 131–140. Zbl0483.60008MR618812
- [83] A. Valette, Les représentations uniformément bornées associées à un arbre réel. Bull. Soc. Math. Belgique42 (1990) 747–760. Zbl0727.43002MR1316222
- [84] H. Wang, Two-point homogeneous spaces. Ann. Math.2 (1952) 177–191. Zbl0048.40503MR47345
- [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).
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