Bernoulli

  • Bernoulli
  • Volume 13, Number 3 (2007), 712-753.

Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes

Arnaud Begyn

Full-text: Open access

Abstract

Cohen, Guyon, Perrin and Pontier have given assumptions under which the second-order quadratic variations of a Gaussian process converge almost surely to a deterministic limit. In this paper we present two new convergence results about these variations: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we apply these results to identify two-parameter fractional Brownian motion and anisotropic fractional Brownian motion.

Article information

Source
Bernoulli, Volume 13, Number 3 (2007), 712-753.

Dates
First available in Project Euclid: 7 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1186503484

Digital Object Identifier
doi:10.3150/07-BEJ5112

Mathematical Reviews number (MathSciNet)
MR2348748

Zentralblatt MATH identifier
1143.60030

Keywords
almost sure convergence central limit theorem fractional processes Gaussian processes generalized quadratic variations

Citation

Begyn, Arnaud. Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. Bernoulli 13 (2007), no. 3, 712--753. doi:10.3150/07-BEJ5112. https://projecteuclid.org/euclid.bj/1186503484


Export citation

References

  • Adler, R.J. (1981). The Geometry of Random Fields. Chichester: Wiley.
  • Ayache, A., Bonami, A. and Estrade, A. (2005). Identification and series decomposition of anisotropic Gaussian fields. In Proceedings of 5th ISAAC Congrees. Catania.
  • Baxter, G. (1956). A strong limit theorem for Gaussian processes. Proc. Amer. Soc. 7 522–527.
  • Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Math. Iboamerica 13 7–18.
  • Bégyn, A. (2005). Quadratic variations along irregular subdivisions for Gaussian processes. Electronic J. Probab. 10 691–717.
  • Biermé, H. (2006). Estimation of anisotropic Gaussian fields through Radon transform. Submitted article. Available at http://www.univ-orleans.fr/mapmo/membres/bierme/recherche/Estimation.pdf.
  • Bingham, H., Goldie, C. and Teugels, J. (1989). Regular Variation. Cambridge: Cambridge Univ. Press.
  • Bonami, A. and Estrade, A. (2003). Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9 215–236.
  • Coeurjolly, J. (2000). Inférence statistique pour les mouvements Brownien fractionnaires et multifractionnaires (French). Ph.D. thesis, Univ. Grenoble I.
  • Cohen, S., Guyon, X., Perrin, O. and Pontier, M. (2005). Singularity functions for fractional processes, and application to fractional Brownian sheet. Ann. Inst. H. Poincaré Probab. Statist. 42 187–205.
  • Dieudonné, J. (1980). Calcul Infinitésimal (French). Paris: Hermann.
  • Hanson, D. and Wright, F. (1971). A bound on tail probabilities for quadratic forms in indenpedent random variables. Ann. Math. Statist. 42 1079–1083.
  • Houdré, C. and Villa, J. (2003). An example of infinite dimensional quasi-helix. Contemp. Math. 336 195–201. Providence, RI: Amer. Math. Soc.
  • Isserlis, L. (1918). On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika 12 134–139.
  • Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 33 407–436.
  • Samorodnitsky, G. and Taqqu, M. (1994). Stable Non-Gaussian Random Processes. London: Chapmann and Hall.
  • Wong, R. (1989). Asymptotic Approximation of Integrals. Boston: Academic Press.