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Consider ψ and ˜ ψ = ψ ˆ f ; by the Lemma ( ˆ m, ˜ ψψ$_{v}$ ) = 

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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$_{v}$ } 

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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 

