Asymptotics for the $L^p$-deviation of the variance estimator under diffusion

We consider a diffusion process $X_t$ smoothed with (small) sampling parameter $\epsilon$. As in Berzin, León and Ortega (2001), we consider a kernel estimate ${\widehat{\alpha }}_{\epsilon }$ with window $h\left(\epsilon \right)$ 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

$$\frac{1}{\sqrt{h}}{\left(\frac{h}{\epsilon }\right)}^{\frac{p}{2}}\left({∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p-𝔼{∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p\right).$$

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