Then Pr(sup Λ ∣ ∣$^{ˆ}$ φ − 1 n − φ − 1 ∣ > ζ$_{7}$ ) ≤ Pr(sup Λ ∣ ∫ s 0 ∑ d k $_{=1}$ˆ κ$_{kn}$ ( t ) dt$_{k}$ − ∫ s 0 ∑ d k $_{=1}$κ$_{k}$ ( t ) dt$_{k}$ ∣ > ln(1 + ˜ B − $^{1}$ζ$_{7}$ )) , and is smaller than 4 ζ$_{2}$ for ζ$_{7}$ = ln(1 + ˜ B − $^{1}$ζ$_{6}$ ) . Since for the continuous functions sup Λ ∣ φ − 1 n ˆ ε$_{1}$$_{n}$ − φ − $^{1}$ε$_{1}$ ∣ < by similar derivations if ζ$_{8}$ < min { ˜ B − $^{1}$ζ$_{1}$ , ζ$_{7}$ (sup | ε$_{1}$ | ) − $^{1}$. ˜ B sup Λ | ε$_{1}$ − ˆ ε$_{1}$$_{n}$ | + sup Λ | ε$_{1}$ | sup Λ ∣ φ − 1 n − φ − $^{1}$∣ ∣ Pr ( sup Λ ∣ ∣$^{ˆ}$ φ − 1 n ˆ ε$_{1}$$_{n}$ − φ − $^{1}$ε$_{1}$ ∣ > ζ$_{8}$ ) < 5 ζ$_{2}$ Λ If Ω is bounded then by Assumption 5 sup Ω ∣ φ − $^{1}$( s ) ∣ is uniformly bounded and since ˆ φ − 1 n converges to φ − 1 in probability uniformly on any compact set inside Ω then also ˆ φ − 1 n is bounded away from zero and then ∣ ∣$^{ˆ}$ φ − 1 n ˆ ε$_{1}$$_{n}$ − φ − $^{1}$ε$_{1}$ ∣ converges in probability to zero on Ω . This concludes the proof of (i) . Step 3. If supp( γ ) is unbounded consider ˜ φ − 1 n defined in (ii). From the proof in step 1 it follows that for large enough N the estimator ˜ φ − 1 n = ˆ φ − 1 n on any compact Λ with arbitrarily high probability and then ˆ ε$_{1}$$_{n}$ ˜ φ − 1 n converges to γ on Λ in probability in S $^{∗}$. Consider an arbitrary set ψ$_{1}$, ..., ψ$_{v}$ ∈ S and the corresponding compact 65