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Article Dans Une Revue Stochastic Processes and their Applications Année : 2008

Assessing the number of mean-square derivatives of a Gaussian process

Résumé

We consider a real Gaussian process $X$ with unknown smoothness $\ro\in\n_{ßte 0}$ where the mean-square derivative $X^{(\ro)}$ is supposed to be H\"{o}lder continuous in quadratic mean. First, from the discrete observations $X(t_1), \dotsc, X(t_n)$, we study reconstruction of $X(t)$, $t\in[0,1]$ with $\widetilde{X}_r(t)$, a piecewise polynomial interpolation of degree $r\ge 1$. We show that the mean-square error of interpolation is a decreasing function of $r$ but becomes stable as soon as $r\ge \ro$. Next, from an interpolation-based empirical criterion, we derive an estimator $\widehat{r}$ of $\ro$ and prove its strong consistency by giving an exponential inequality for $P(\widehat{r}\not=\ro)$. Finally, we prove the strong consistency of $\widetilde{X}_{\widehat{r}}(t)$ with an almost optimal rate.
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Dates et versions

hal-00019204 , version 1 (17-02-2006)
hal-00019204 , version 2 (20-02-2006)

Identifiants

  • HAL Id : hal-00019204 , version 2

Citer

Delphine Blanke, Céline Vial. Assessing the number of mean-square derivatives of a Gaussian process. Stochastic Processes and their Applications, 2008, 118 (10), pp 1852-1869. ⟨hal-00019204v2⟩
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