a similar assumption for G holds, or ¯ G is the indicator function, the band- width parameter h = cn − $^{α}$, where α < 1 4 and Assumption 2 holds. Then for a random sample { ( x$_{i}$, y$_{i}$ ) } n i =1 the estimator ˆ F$_{y}$$_{|}$$_{x}$ ( x, y ) as a generalized random function on D ( W ) converges to the conditional distribution generalized func- tion F$_{y}$$_{|}$$_{x}$ defined by (11) at the rate n − 1 2 ; the limit process for √$_{n}$$_{(}$ ˆ F$_{y}$$_{|}$$_{x}$ − F$_{y}$$_{|}$$_{x}$ ) on D ( W ) is given by a ψ ∈ D ( W ) indexed random functional, Q$_{y}$$_{|}$$_{x}$ with ( Q$_{y}$$_{|}$$_{x}$, ψ ) = where U$_{x}$, U$_{xy}$ are Brownian bridge processes with dimension d$_{x}$, d$_{y}$ + d$_{x}$, correspondingly; as a generalized random process the limit process Q$_{y}$$_{|}$$_{x}$ of √$_{n}$$_{(}$ ˆ F$_{y}$$_{|}$$_{x}$ − F$_{y}$$_{|}$$_{x}$ ) is Gaussian with mean functional zero and covariance bilin- ear functional C, given for any ψ$_{1}$, ψ$_{2}$ by This result is general in that the root-n convergence holds here regardless of whether the marginal density exists. If it does exist the result could be restated for conditional distribution as a generalized function on D$_{0}$ ( R d$_{x}$ ) by (15) . Remark 3. Sometimes for a singular distribution the kernel estimator ˆ f$_{x}$ ( x ) diverges at a specific rate, as e.g. in Lu (1999) where at points x 24 ( C, ( ψ$_{1}$, ψ$_{2}$ )) = cov [( Q$_{y}$$_{|}$$_{x}$, ψ$_{1}$ ) , ( Q$_{y}$$_{|}$$_{x}$, ψ$_{2}$ ) . ( − 1) d$_{x}$ [ ∫ F$_{xy}$ ( ∂ $^{d$_{x}$}$∂$^{d$_{x}$}$ψ )( F$_{x}$ ) U$_{x}$dF$_{x}$ + ∫ F$_{xy}$ ( ∂ $^{d$_{x}$}$ψ )( F$_{x}$ ) dU$_{x}$ + ∫ ( ∂ $^{d$_{x}$}$ψ )( F$_{x}$ ) U$_{xy}$dF$_{x}$ ] , Proof. See Appendix.