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Asymptotics for the \(L^p\)-deviation of the variance estimator under diffusion. (English) Zbl 1186.60080

Summary: We consider a diffusion process \(X_t\) smoothed with (small) sampling parameter \(\varepsilon \). As in C. Berzin-Joseph, J.R. León and J. Ortega [Stoch. Proc. Appl. 92, 11–30 (2001; Zbl 1047.60082)], we consider a kernel estimate \(\widehat{\alpha }_{\varepsilon }\) with window \(h(\varepsilon )\) of a function \(\alpha \) of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the \(L^p\) deviations such as \[ \frac{1}{\sqrt{h}}\left(\frac{h}{\varepsilon }\right)^{\frac{p}{2}}\left( \left\| \widehat{\alpha }_{\varepsilon }-{\alpha }\right\| _p^p- \mathbb{E}\left\| \widehat{\alpha }_{\varepsilon }-{\alpha }\right\| _p^p \right). \]

MSC:

60J65 Brownian motion
62M05 Markov processes: estimation; hidden Markov models
60F05 Central limit and other weak theorems
60F25 \(L^p\)-limit theorems
60H05 Stochastic integrals
62M02 Markov processes: hypothesis testing

Citations:

Zbl 1047.60082
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References:

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