( 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}$, note that by construction these random gener- alized functions are infinitely differentiable since by the assumption on the kernel K the support of ˆ w$_{·}$$_{n}$ is bounded and thus the Fourier transform is differentiable. The Theorem below establishes consistency of plug-in estima- tors. Theorem 6 If Assumptions 4-7 are satisfied, then (i) if supp( γ ) is bounded the plug-in estimator ˆ γ$_{n}$ = [ exp( − ∫ s 0 d ∑ k =1 ˆ ε − 1 1 $_{n}$( t )((ˆ ε$_{1}$$_{n}$ ( t )) ′ k − i ˆ ε$_{2}$$_{kn}$ ( t )) dt$_{k}$ ] ˆ ε$_{1}$$_{n}$ ˜ φ − 1 = ˜ φ − 1 n ( s ) = [ exp( − ∫ s 0 d ∑ k =1 ˆ ε − 1 1 $_{n}$( t )((ˆ ε$_{1}$$_{n}$ ( t )) ′ k − i ˆ ε$_{2}$$_{kn}$ ( t )) dt$_{k}$ ] , 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}$$_{n}$ with if and ∣ [ exp( − ∫ s 0 d ∑ k =1 ˆ ε − 1 1 $_{n}$( t )((ˆ ε$_{1}$$_{n}$ ( t )) ′ k − i ˆ ε$_{2}$$_{kn}$ ( t )) dt$_{k}$ ]∣ ∣ < V ( 1 + s $^{2}$) m V ( 1 + s $^{2}$) m 59