content_image/309435.jpg content_image/309436.jpg content_image/309437.jpg
and by the Cramer-Wold theorem convergence to a limit Gaussian process for ˆ S − 1 / 2 n − → η nh and thus for S − 1 / 2 − → η hn follows. ■ 

## Proof of Theorem 3. 

Since for a smooth kernel ˆ F ( x, y ) ∈ Φ$_{c}$ by the Lemma the value of the functional for ψ ∈ D (0 , 1) d$_{x}$ , ( ˆ F$_{y}$$_{|}$$_{x}$, ψ ) is the same as ( ˆ F$_{y}$$_{|}$$_{x}$, ˜ ψ ) , with the latter defined by (13) where ˜ ψ = ˆ f$_{x}$ψ ( ˆ F$_{x}$ ) . Thus for any ψ ∈ D (0 , 1) : 

<img src='content_image/309435.jpg'>

More concisely it is ( ˆ F$_{y}$$_{|}$$_{x}$, ψ ) = 

<img src='content_image/309436.jpg'>

Here ”hat” indicates empirical distribution function and ”tilde” the kernel estimated distribution function. By standard arguments the smooth kernel introduces a bias; by the usual expansions using differentiability of ψ we get that for the second order kernel 

<img src='content_image/309437.jpg'>

