Then Pr(inf A | ˆ ε$_{1}$$_{n}$ | < ζ$_{1}$ ) < Pr(inf A ∣ ˆ ε$_{1}$$_{n}$ − ( ε$_{g}$ )$_{n}$ ∣ < ζ$_{1}$ ) < ζ$_{2}$ ; and (23) also holds. Then with probability approaching 1 on compact Λ the continuous ran- dom functions ˆ κ$_{kn}$ = ˆ ε − 1 1 $_{n}$((ˆ ε$_{1}$$_{n}$ ) ′ k − i ˆ ε$_{2}$$_{kn}$ ) , k = 1 , ..., d are in a bounded set in S $^{∗}$, and thus ˆ φ − 1 n ( s ) = exp( − ∫ s 0 ∑ d k $_{=1}$ˆ κ$_{k}$ ( t ) dt$_{k}$ ) is also in a bounded set, and thus with probability approaching 1 satisfies the condition for well-posedness of Theorem 5. Then the estimators ˆ γ$_{n}$ = ˆ φ − 1 n ε$_{1}$ are consistent for γ in S ∗ on the compact set Λ where inf Λ | ε$_{1}$$_{L}$$_{1}$ | > 0 . Since γ$_{L}$$_{1}$ is a continuous function its support is an open set. By Assump- tion 6(c’) the compact support of γ$_{g}$ is contained inside the open support of γ$_{L}$$_{1}$. If inf supp( γ$_{l}$$_{1}$ $_{)}$| ε$_{1}$$_{L}$$_{1}$ | > ζ$_{1}$ > 0 this concludes the proof of (i). Step 2. Otherwise consider the set ¯ Λ = ∪ Λ$_{gn}$; and the open set Ω =supp( γ ) $_{⧹}$¯ Λ; then ˆ ε$_{1}$$_{n}$ converge to ε$_{1}$ that is continuous on that set. Of course under As- sumption 6(c) Ω = supp ( γ ) . Generally Ω is a union of open connected sets and we can proceed by considering each component in Ω . Without loss of generality such a component can be assumed to be a connected open set containing zero as an interior point, since if it does not contain zero by a shift (which is a continuous operation in S $^{∗}$) of an arbitrary interior point into zero this can be attained. All the proofs that follow apply to an open set that will be denoted Ω 62