We consider a diffusion process Xt smoothed with (small)
sampling parameter ε. As in Berzin, León and Ortega
(2001), we consider a kernel estimate
$\widehat{\alpha}_{\varepsilon}$ with window h(ε) of a
function α of its variance. In order to exhibit global
tests of hypothesis, we derive here central limit theorems for
the Lp deviations such as
\[
\frac1{\sqrt{h}}\left(\frac{h}\varepsilon\right)^{\frac{p}2}\left(
\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p-
\mbox{I E}\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p
\right).
\]