B Additional Proofs B.1 Sketch of Proof of Theorem 4 The proof of Theorem 4 is very similar to that of Theorem 1. We only sketch it and point out the places where they differ. Sketch of Proof of Theorem 4. Obviously (9) implies that R ( · ) is uniformly continuous on Π with respect to ‖·‖ . Assumption (8) then shows that the implication of Lemma A.3 also holds in our context, i.e. , that inf$_{π}$$_{∈}$$_{Π}$ R ( π ) = 0 and that Π ⋆ is not empty. As R ( · ) is uniformly continous and the implication of Lemma A.3 holds, (CA) from Lemma A.4 also holds. Furthermore, (10) yields Lemma A.5, for which Corollary A.6 remains a valid corollary. We will not make use of Lemma A.7: we will instead use directly (12). We now have the tools needed to modify the proof of Theorem 1. Using the results we have obtained thus far, and replacing (6) by (13) and (5) by (11), we see that (A.13) from the proof of Theorem 1 can be replaced by We tighten this result to R $^{(}$̂ π$_{n}$ ) = o$_{P}$ ( r − 1 n ) as in the proof of Theorem 1. In particular, nearly identical arguments to those used in that proof show that By (12) and the continuous mapping theorem [13, Theorem 1.3.6], sup$_{f}$$_{∈F}$ | ˜ G $_{n}$f | = O$_{P}$ (1). Hence, the leading term in the final inequality is O$_{P}$ ( r − 1 n ), where this term does not depend on s . The final term also does not depend on s . By (9), the middle term above goes to zero as s ↓ 0, where this convergence is uniform in both π ⋆ ∈ Π ⋆ and π ⋆ s ∈ B$_{Π}$ ( π $^{⋆}$, s ). Thus, R $^{(}$̂ π$_{n}$ ) = O$_{P}$ ( r − 1 n ). We tighten this result to R π$_{n}$ ) = o$_{P}$ ( r − 1 n ) as in the proof of Theorem 1. In particular, 0 ≤ R $^{(}$̂ π$_{n}$ ) ≲ r − 1 n [ ˜ G $_{n}$f$_{̂}$ π$_{n}$ − ˜ G $_{n}$f$_{π}$ ⋆ s ] + [ V ( π $^{⋆}$) − V ( π ⋆ $_{s}$)] + o$_{P}$ ( r − 1 n ) (B.16) ≤ 2 r − 1 n sup f ∈F | ˜ G $_{n}$f | + [ V ( π $^{⋆}$) − V ( π ⋆ $_{s}$)] + o$_{P}$ ( r − 1 n ) . 0 ≤ r$_{n}$ R $^{(}$̂ π$_{n}$ ) ≤ g ( ˜ G $_{n}$, R $^{(}$̂ π$_{n}$ ) + t$_{n}$ ) + o$_{P}$ (1) ≤ h ( ˜ G $_{n}$, R $^{(}$̂ π$_{n}$ ) + t$_{n}$ ) + o$_{P}$ (1) . 31