otherwise, is such that it exists with probability approaching 1 and F t − $^{1}$(ˆ γ$_{n}$ ) − g →$_{p}$ 0 in S $^{∗}$. 

Proof. The proof consists of the following steps. 

Step 1 considers a compact set inside the support of γ and shows that on such a set consistency in S ∗ follows. 

Step 2 examines the continuous part of γ, γ$_{L}$$_{1}$, and convergence is prob- ability of estimators defined on compact sets that exclude a set containing the support of the singular part. 

Combining the results of Step 1 and 2 concludes the proof of (i). 

Step 3 considers the case of unbounded support with ˜ φ − 1 n defined in (ii). On any compact set with this estimator replacing ˆ φ − 1 n the results in Steps 1 and 2 hold so consistency on a bounded set obtains. Consistency on the unbounded support is shown in topology of S ∗ by selecting for any set of ψ$_{1}$, ...ψ$_{v}$ from S the corresponding compact set defined in Lemma 3 to bound all the functionals in probability outside of the compact set to prove (ii). 

Next, the details are provided. 

Step 1. First, ˆ ε$_{1}$$_{n}$ ; this is a sequence of continuous functions that converge in probability in S ∗ by Lemma 3. Fourier transforms of functions in L$_{1}$ are uniformly continuous on compact sets. If Assumption 6(c) holds convergence is to the continuous function ε$_{1}$ . Then ˆ ε$_{1}$$_{n}$ converge in probability pointwise and uniformly on the compact set Λ . Then for any 0 < ζ$_{1}$, ζ$_{2}$ < 1 we can find 

