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Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. (English) Zbl 1143.60030

The author considers second-order quadratic variations
\[ \sum_{k=1}^{n-1}[X_{(k+1)/n}+X_{(k-1)/n}-2X_{k/n}]^2, \qquad n\in\mathbf N, \]
of a Gaussian process \(X=\{X_t; t\in[0,1]\}.\) From S. Cohen, X. Guyon, O. Perrin and M. Pontier [Ann. Inst. Henri Poincaré, Probab. Stat. 42, No. 2, 187–205 (2006; Zbl 1095.60011)] we know that it converges to a deterministic limit under convenient conditions on the covariance function of the process. The author sharpens this result: he shows that if an asymptotic expansion of the covariance function is known, then one can get a deterministic asymptotic expansion of the second-order quadratic variation. Then the author establishes centered limit theorems related to the previous results. As applications, the results are applied to estimate the parameters of some fractional processes (i.e., processes whose properties are close to those of fractional Brownian motion). To explain how to use the results, first the author applies them to the fractional Brownian motion, even though the consequences are not new in this case. Then the two-parameter fractional Brownian motion \(B^{H,K}=\{B_t^{H,K};t\in\mathbb{R}\}\) is considered, which is defined by, for \(H\in(0,1)\) and \(K\in(0,1],\) as the unique continuous centered Gaussian process with covariance function \[ \frac{1}{2^K}((s^{2H}+t^{2H})^K-| s-t|^{2HK}), \qquad s,t\in\mathbb{R}. \] Besides studying asymptotic properties of the second-order quadratic variations of \(B^{H,K},\) the author constructs strongly consistent and asymptotically normal estimators of \(H K,\) \(H\) and \(K.\) Finally, one can find applications of the results to anisotropic fractional Brownian motions.

MSC:

60G15 Gaussian processes
60F05 Central limit and other weak theorems
60F10 Large deviations

Citations:

Zbl 1095.60011
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References:

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