Consider ψ and ˜ ψ = ψ ˆ f ; by the Lemma ( ˆ m, ˜ ψψ$_{v}$ ) = Assumption 5. The conditional variance σ $^{2}$( x ) = E$_{y}$$_{|}$$_{x}$y 2 defines a generalized function on D ( W ). Assumption 5 implies that for any ψ ∈ D ( W ) the value of the func- tional ( σ $^{2}$, ψ ) = ∫ σ $^{2}$( x ) ψ ( F$_{x}$ ( x )) dF$_{x}$ ( x ) is always bounded; this is reqired to bound the variance for the limit process. By (21) for a partition of unity, { ψ$_{v}$ } 30 Theorem 4. Suppose that Assumptions 1-5 hold, the bandwidth param- eter h = cn − $^{α}$, where α < 1 $_{4}$. Then the estimator ˆ m ( x ) for a random sample { ( x$_{i}$, y$_{i}$ ) } n i =1 as a generalized random function on D ( W ) converges at the rate n − 1 2 to the generalized function m that provides (22) ; the limit process for √$_{n}$$_{( ˆ}$ m − m ) on D ( W ) is given by a ψ ∈ D ( W ) indexed random functional ( σ $^{2}$, ψ ) = Σ$_{v}$ ( − 1) d$_{x}$ +1 ∫ ∫ F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ ( F$_{x}$ ( x )) ( y $^{2}$ψ$_{v}$ ( y ) $^{′}$dF$_{x}$ ( x ) dy. ( − 1) d$_{x}$ +1 ∫ ∫ ˆ F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ ( ˆ F$_{x}$ ( x ) ) d dy [ yψ$_{v}$ ( y )] d ( ˆ F$_{x}$ ( x ) ) dy (23) = ( − 1) d$_{x}$ +1 ˆ F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ ˆ F$_{x}$ ( x ) [ yψ ′ $_{v}$( y ) + ψ$_{v}$ ( y )] d ( ˆ F$_{x}$ ( x ) ) dy.