content_image/302358.jpg content_image/302357.jpg
Q$_{m}$ with ( Q$_{m}$, ψ ) = 

<img src='content_image/302358.jpg'>

where U$_{x}$, U$_{x,y}$ are Brownian bridge processes with dimension d$_{x}$, d$_{x}$ + 1 , correspondingly; as a generalized random process the limit process Q$_{m}$ of √$_{n}$$_{( ˆ}$ m − m ) is Gaussian with mean functional zero and covariance bilinear functional C, given for any ψ$_{1}$, ψ$_{2}$ by 

Proof. See Appendix. 

<img src='content_image/302357.jpg'>

Similarly to the kernel estimator for the conditional distribution the con- ditional mean estimator converges at parametric rate as a functional on D ( W ) for any distribution. When a positive conditioning density exists it is possible to represent the conditional mean as a functional on D ( R d$_{x}$ ) , by the same arguments as in the Lemma. In the case of Remark 3 a similar rescaling provides a faster convergence rate for the estimator considered as a functional on D ( R d$_{x}$ ) . 

