content_image/310918.jpg content_image/310919.jpg content_image/310923.jpg content_image/310924.jpg
( F t ( ˜ w$_{n}$ − w ) , ψ ) = ( ˆ w$_{n}$ − ˆ w, F t ( ψ ))) . Thus convergence in probability in S ∗ of the estimators of the functions ˆ w, and also of the Fourier transforms ε is established. ■ 

Denote F t ( ˜ w$_{·}$$_{n}$ ) by ˆ ε$_{·}$$_{n}$, note that by construction these random gener- alized functions are infinitely differentiable since by the assumption on the kernel K the support of ˆ w$_{·}$$_{n}$ is bounded and thus the Fourier transform is differentiable. The Theorem below establishes consistency of plug-in estima- tors. 

## Theorem 6 If Assumptions 4-7 are satisfied, then 

(i) if supp( γ ) is bounded the plug-in estimator 

<img src='content_image/310918.jpg'>

is such that it exists with probability going to 1 and F t − $^{1}$(ˆ γ$_{n}$ ) − g →$_{p}$ 0 in S $^{∗}$; (ii) generally the estimator ˆ γ$_{n}$ = ˜ φ − $^{1}$ˆ ε$_{1}$$_{n}$ with 

if 

and 

<img src='content_image/310919.jpg'>

<img src='content_image/310923.jpg'>

<img src='content_image/310924.jpg'>

