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Functional limit theorems for generalized quadratic variations of Gaussian processes

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  • Bégyn, Arnaud

Abstract

In this paper, we establish functional convergence theorems for second order quadratic variations of Gaussian processes which admit a singularity function. First, we prove a functional almost sure convergence theorem, and a functional central limit theorem, for the process of second order quadratic variations, and we illustrate these results with the example of the fractional Brownian sheet (FBS). Second, we do the same study for the process of localized second order quadratic variations, and we apply the results to the multifractional Brownian motion (MBM).

Suggested Citation

  • Bégyn, Arnaud, 2007. "Functional limit theorems for generalized quadratic variations of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1848-1869, December.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:12:p:1848-1869
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    References listed on IDEAS

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    1. Benassi, Albert & Cohen, Serge & Istas, Jacques, 1998. "Identifying the multifractional function of a Gaussian process," Statistics & Probability Letters, Elsevier, vol. 39(4), pages 337-345, August.
    2. Benassi, Albert & Cohen, Serge & Istas, Jacques & Jaffard, Stéphane, 1998. "Identification of filtered white noises," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 31-49, June.
    3. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    4. Perrin, Olivier, 1999. "Quadratic variation for Gaussian processes and application to time deformation," Stochastic Processes and their Applications, Elsevier, vol. 82(2), pages 293-305, August.
    5. Jean-François Coeurjolly, 2001. "Estimating the Parameters of a Fractional Brownian Motion by Discrete Variations of its Sample Paths," Statistical Inference for Stochastic Processes, Springer, vol. 4(2), pages 199-227, May.
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    Cited by:

    1. Bardet, Jean-Marc & Surgailis, Donatas, 2013. "Nonparametric estimation of the local Hurst function of multifractional Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 1004-1045.

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