Q$_{m}$ with ( Q$_{m}$, ψ ) = 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}$$_{( ˆ}$ m − m ) is Gaussian with mean functional zero and covariance bilinear functional C, given for any ψ$_{1}$, ψ$_{2}$ by Proof. See Appendix. Similarly to the kernel estimator for the conditional distribution the con- ditional mean estimator converges at parametric rate as a functional on D ( W ) for any distribution. When a positive conditioning density exists it is possible to represent the conditional mean as a functional on D ( R d$_{x}$ ) , by the same arguments as in the Lemma. In the case of Remark 3 a similar rescaling provides a faster convergence rate for the estimator considered as a functional on D ( R d$_{x}$ ) . 31 ( C, ( ψ$_{1}$, ψ$_{2}$ )) = cov [( Q$_{m}$, ψ$_{1}$ ) , ( Q$_{m}$, ψ$_{2}$ ) . Σ$_{v}$( − 1) d$_{x}$ +1 ∫ ... { ∫ U$_{x,y}$∂ $^{d$_{x}$}$ψ ( F$_{x}$ ( x )) dF$_{x}$ ( x ) + ∫ F$_{x,y}$ ( x, y ) ( ∂ d$_{x}$ ) $^{2}$ψ ( F$_{x}$ ( x )) U$_{x}$dF$_{x}$ ( x ) + F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ ( F$_{x}$ ( x )) dU$_{x}$ } [ yψ $^{′}$$_{v}$( y ) + ψ$_{v}$ ( y )] dy$_{1}$...dy$_{d}$$_{y}$,