ASYMPTOTICS FOR THE Lp-DEVIATION OF THE VARIANCE ESTIMATOR UNDER DIFFUSION

We consider a diffusion process Xt smoothed with (small) sampling parameter ε. As in Berzin, León and Ortega (2001), we consider a kernel estimate α̂ε 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 L deviations such as 1 √ h ( h ε ) p 2 ( ‖α̂ε − α‖pp − IE ‖α̂ε − α‖pp ) . Mathematics Subject Classification. 60F05, 60F25, 60J60, 60H05, 62M02, 62M05. Received October 22, 2002. Revised October 21, 2003.