Issue
ESAIM: PS
Volume 16, 2012
Special Issue: Spring School Mons Random differential equations and Gaussian fields
Page(s) 222 - 276
DOI https://doi.org/10.1051/ps/2011106
Published online 11 July 2012
  1. P. Abry, P. Gonçalvès and P. Flandrin, Wavelets, spectrum analysis and 1 / f processes. Lect. Note Stat. 103 (1995) 15–29. [CrossRef] [Google Scholar]
  2. A. Ayache and J. Lévy-Vehel, The Multifractional Brownian motion. Stat. Inference Stoch. Process. 1 (2000) 7–18. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  7. B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s Property (T). Cambridge University Press (2008). [Google Scholar]
  8. A. Benassi, S. Jaffard and D. Roux, Gaussian processes and Pseudodifferential Elliptic operators. Revista Mathematica Iberoam. 13 (1997) 19–90. [CrossRef] [MathSciNet] [Google Scholar]
  9. A. Benassi, S. Cohen and J. Istas, Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett. 39 (1998) 337–345. [CrossRef] [Google Scholar]
  10. A. Benassi, S. Cohen, J. Istas and S. Jaffard, Identification of filtered white noises. Stoc. Proc. Appl. 75 (1998) 31–49. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  12. A. Benassi, S. Cohen and J. Istas, Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8 (2002) 97–115. [MathSciNet] [Google Scholar]
  13. A. Benassi, S. Cohen and J. Istas, On roughness indices for fractional fields. Bernoulli 10 (2004) 357–373. [CrossRef] [MathSciNet] [Google Scholar]
  14. A. Begyn, Quadratic variations along irregular subdivisions for Gaussian processes. Electron. J. Probab. 10 (2005) 691–717. [CrossRef] [MathSciNet] [Google Scholar]
  15. A. Begyn, Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. Bernoulli 13 (2007) 712–753. [CrossRef] [MathSciNet] [Google Scholar]
  16. A. Begyn, Functional limit theorems for generalized quadratic variations of Gaussian processes. Stoc. Proc. Appl. 117 (2007) 1848–1869. [CrossRef] [Google Scholar]
  17. C. Berzin and J. Leon, Estimating the Hurst parameter. Stat. Inference Stock. Process. 10 (2007) 49–73. [CrossRef] [Google Scholar]
  18. A. Bonami and A. Estrade, Anisotropic analysis of Gaussian models. J. Fourier Anal. Appl. 9 (2004) 215–236. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  20. J. Bretagnolle, D. Dacunha-Castelle and J.-L. Krivine, Lois stables et espaces Lp. Ann. Inst. Henri Poincaré 2 (1969) 231–259. [Google Scholar]
  21. A. Brouste, J. Istas and S. Lambert-Lacroix, On fractional Gaussian random fields simulation. J. Stat. Soft. 1 (2007) 1–23. [Google Scholar]
  22. A. Brouste, J. Istas and S. Lambert-Lacroix, On simulation of fractional Brownian motion indexed by a manifold. J. Stat. Soft. 36 (2010). [Google Scholar]
  23. N. Chentsov, Lévy’s Brownian motion of several parameters and generalized white noise. Theory Probab. Appl. 2 (1957) 265–266. [CrossRef] [Google Scholar]
  24. J.-F. Coeurjolly, Simulation and identification of the fractional Brownian motion : a bibliographical and comparative study. J. Stat. Software 5 (2000) 1–53. [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  26. J.-F. Coeurjolly, Identification of multifractional Brownian motion. Bernoulli 11 (2005) 987–1008. [CrossRef] [MathSciNet] [Google Scholar]
  27. J.-F. Coeurjolly, Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles. Ann. Statist. 36 (2008) 1404–1434. [CrossRef] [MathSciNet] [Google Scholar]
  28. J.-F. Coeurjolly and J. Istas, Cramer-Rao bounds for fractional Brownian motions. Stat. Probab. Lett. 53 (2001) 435–447. [CrossRef] [Google Scholar]
  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). [Google Scholar]
  30. S. Cohen and J. Istas, An universal estimator of local self-similarity. Preprint (2006). [Google Scholar]
  31. S. Cohen and J. Istas, Fractional fields : Modelling and statistical applications (Submitted). [Google Scholar]
  32. S. Cohen and M. Lifshits, Stationary Gaussian random fields on hyperbolic spaces and Euclidean spheres. To appear in ESAIM : PS. [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  34. D. Dacunha-Castelle and M. Duflo, Probabilités et Statistiques tome 2. Masson, Paris (1983). [Google Scholar]
  35. R. Dalhaus, Efficient parameter estimation for self-similar processes. Ann. Statist. 17 (1989) 1749–1766. [CrossRef] [MathSciNet] [Google Scholar]
  36. I. Daubechies, Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41 (1988) 909–996. [CrossRef] [MathSciNet] [Google Scholar]
  37. S. Dégerine and S. Lambert-Lacroix, Partial autocorrelation function of a nonstationary time series. J. Multiv. Anal. (2003) 46–59. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  39. A. Dress, V. Moulton and W. Terhalle, T-theory : An overview, Eur. J. Comb. 17 (1996) 161–175. [CrossRef] [Google Scholar]
  40. A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher transcendental functions (Bateman manuscript project). McGraw-Hill 2 (1953) [Google Scholar]
  41. K. Falconer, Tangent fields and the local structure of random fields. J. Theor. Probab. 15 (2002) 731–750. [CrossRef] [MathSciNet] [Google Scholar]
  42. K. Falconer, The local structure of random processes. J. Lond. Math. Soc. 67 (2003) 657–672. [CrossRef] [MathSciNet] [Google Scholar]
  43. J. Faraut, Fonction brownienne sur une variété riemannienne. Séminaire de probabilités de Strasbourg 7 (1973) 61–76. [Google Scholar]
  44. J. Faraut and H. Harzallah, Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier 24 (1974) 171–217. [CrossRef] [MathSciNet] [Google Scholar]
  45. S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, 2nd edition. Springer-Verlag (1993). [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  48. S. Helgason, Differential Geometry and Symmetric spaces. Academic Press (1962). [Google Scholar]
  49. E. Herbin and E. Merzbach, A set-indexed fractional Brownian motion. J. Theor. Probab. 19 (2006) 337–364. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  51. J. Istas, Spherical and hyperbolic fractional Brownian motion. Electron. Comm. Probab. 10 (2005) 254–262. [CrossRef] [MathSciNet] [Google Scholar]
  52. J. Istas, On fractional stable fields indexed by metric spaces. Electron. Comm. Probab. 11 (2006) 242–251. [CrossRef] [MathSciNet] [Google Scholar]
  53. J. Istas, Karhunen-Loève expansion of spherical fractional Brownian motions. Stat. Probab. Lett. 76 (2006) 1578–1583. [CrossRef] [Google Scholar]
  54. J. Istas, Quadratic variations of spherical fractional Brownian motions, Stoc. Proc. Appl. 117 (2007) 476–486. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  56. J. Istas and C. Lacaux, On locally self-similar fractional random fields indexed by a manifold. preprint. [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  60. A. Koldobsky, Schoenberg’s problem on positive definite functions. Algebra Anal. 3 (1991) 78–85. [Google Scholar]
  61. A. Koldobsky and Y. Lonke, A short proof of Schoenberg’s conjecture on positive definite functions. Bull. Lond. Math. Soc. (1999) 693–699. [CrossRef] [Google Scholar]
  62. A. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertsche Raum (German). C. R. (Dokl.) Acad. Sci. URSS 26 (1940) 115–118. [Google Scholar]
  63. C. Lacaux, Real harmonizable multifractional Lévy motions. Ann. Inst. Henri Poincaré 40 (2004) 259–277. [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  65. P. Lévy, Processus stochastiques et mouvement Brownien. Gauthier-Vilars (1965). [Google Scholar]
  66. T. Lindstrom, Fractional Brownian fields as integrals of white noise. Bull. Lond. Math. Soc. 25 (1993) 83–88. [CrossRef] [Google Scholar]
  67. M. Maejima, A remark on self-similar processes with stationary increments. Can. J. Stat. 14 (1986) 81–82. [CrossRef] [Google Scholar]
  68. B.B. Mandelbrot and J.W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968) 422–437. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  69. R. Peltier and J. Lévy-Vehel, Multifractional Brownian motion : definition and preliminary results. Rapport de recherche de l’INRIA 2645 (1996). [Google Scholar]
  70. P. Petersen, Riemannian Geometry. Graduate Texts in Mathematics, Springer (1998). [Google Scholar]
  71. E. Rafajlowicz, Testing (non-)existence of input-output relationships by estimating fractal dimensions. IEEE Trans. Signal Process. 52 (2004) 3151–3159. [CrossRef] [Google Scholar]
  72. G. Robertson, Crofton formulae and geodesic distance in hyperbolic spaces. J. Lie Theory 8 (1998) 163–172. [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  74. W. Rudin, Fourier analysis on groups. Wiley (1962). [Google Scholar]
  75. G. Samorodnitsky, Long memory and self-similar processes. Annales de la Faculté des Sciences Toulouse 15 (2006) 107–123. [CrossRef] [MathSciNet] [Google Scholar]
  76. G. Samorodnitsky and M. Taqqu, Stable non-Gaussian random processes : stochastic models with infinite variance. Chapman & Hall, New York (1994). [Google Scholar]
  77. I. Schönberg, Metric spaces and positive definite functions. Ann. Math. 39 (1938) 811–841. [CrossRef] [MathSciNet] [Google Scholar]
  78. R. Seeley, Spherical harmonics. Am. Math. Mon. 73 (1966) 115–121. [CrossRef] [MathSciNet] [Google Scholar]
  79. S. Stoev and M. Taqqu, Stochastic properties of the linear multifractional stable motion. Adv. Appl. Prob. 36 (2004) 1085–1115. [CrossRef] [Google Scholar]
  80. G. Szego, Orthogonal Polynomials, 4th edition, in Amer. Math. Soc. Providence, RI (1975). [Google Scholar]
  81. S. Takenaka, Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123 (1991) 1–12. [MathSciNet] [Google Scholar]
  82. S. Takenaka, I. Kubo and H. Urakawa, Brownian motion parametrized with metric space of constant curvature. Nagoya Math. J. 82 (1981) 131–140. [MathSciNet] [Google Scholar]
  83. A. Valette, Les représentations uniformément bornées associées à un arbre réel. Bull. Soc. Math. Belgique 42 (1990) 747–760. [Google Scholar]
  84. H. Wang, Two-point homogeneous spaces. Ann. Math. 2 (1952) 177–191. [CrossRef] [Google Scholar]
  85. A. Yaglom, Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Probab. Appl. 2 (1957) 273–320. [CrossRef] [Google Scholar]
  86. A. Zaanen, Linear Anal. North Holland Publishing Co (1960). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.