that is connected and contains zero as an interior point. Consider a compact set Λ ⊂ Ω where inf Λ | ε$_{1}$ | > 0 . It follows from the proof in step 1 under continuity of ε$_{1}$ that Pr ( sup Λ ∣ ˆ ε − 1 1 n − ε − 1 1 ∣ > ζ$_{1}$ ) = Pr ( sup Λ ∣ ( ˆ ε − 1 1 $_{n}$( ε$_{1}$ − ˆ ε$_{1}$$_{n}$ ) ε − 1 1 ∣ > ζ$_{1}$ ) ≤ Pr ( sup Λ ∣ ( ˆ ε − 1 1 $_{n}$( ε$_{1}$ − ˆ ε$_{1}$$_{n}$ )∣ ∣ > ζ 2 1 ) ≤ Pr ( sup Λ ∣ ˆ ε − 1 1 n ∣ > ζ − 1 1 , sup Λ ∣ ˆ ε − 1 1 $_{n}$( ε$_{1}$ − ˆ ε$_{1}$$_{n}$ ) ∣ > ζ 2 1 ) + Pr ( sup Λ ∣ ˆ ε − 1 1 n ∣ ≤ ζ − 1 1 , sup Λ ∣ ˆ ε − 1 1 $_{n}$( ε$_{1}$ − ˆ ε$_{1}$$_{n}$ ) ∣ > ζ 2 1 ) ≤ Pr ( sup Λ ∣ ˆ ε − 1 1 n ∣ > ζ − 1 1 ) + Pr ( sup Λ | ε$_{1}$ − ˆ ε$_{1}$$_{n}$ | > ζ 3 1 ) ≤ 2 ζ$_{2}$. Theorem 3(b) implies that φ − $^{1}$( s ) = exp( − ∫ s 0 ∑ d k $_{=1}$κ$_{k}$ ( t ) dt$_{k}$ ) , where κ$_{k}$ is the unique continuous function that solves ε$_{1}$ κ − (( ε$_{1}$ ) ′ k − iε$_{2}$$_{k}$ ) = 0 in S ∗ on Ω . By Assumptions 6(a-b) for φ , ε$_{1}$φ − 1 exists in S $^{∗}$and g = F t − $^{1}$( ε$_{1}$φ − $^{1}$) in S $^{∗}$. By Assumption 5a ε$_{1}$ = γφ is non-zero on supp( γ ). Consider the estimator function ˆ κ$_{kn}$ = ˆ ε − 1 1 $_{n}$((ˆ ε$_{1}$$_{n}$ ) ′ k − i ˆ ε$_{2}$$_{kn}$ ) on compact Λ ⊂ Ω; the function ε$_{1}$ is continuous there and it follows that ( ε$_{1}$ ) ′ k − iε$_{2}$$_{k}$ = ε$_{1}$ κ$_{k}$ is continuous. The sequences of random functions ((ˆ ε$_{1}$$_{n}$ ) $^{′}$$_{k}$− i ˆ ε$_{2}$$_{kn}$ ) converge in probability uniformly on Λ in S ∗ to the continuous function (( ε$_{1}$ ) ′ k − iε$_{k}$$_{2}$ ) . Define ¯ B = 63