as a set of constant functions with a uniformly bounded sup-norm) is also Donsker, [13, Example 1.10.7] yields that { ˜ f$_{π}$ − λ : π ∈ Π , λ ∈ Λ } is Donsker, and so is its subset F . Proof of Theorem 1. In this proof, we make the dependence of ̂ π on n explicit by writing ̂ π$_{n}$ . By Lemma A.7, Π Donsker implies F Donsker. Consider the empirical process G n as the element of ℓ $^{∞}$( F ) characterized by G $_{n}$f ≡ n 1 / $^{2}$( P$_{n}$ − P ) f for every f ∈ F . We let G P ∈ ℓ $^{∞}$( F ) denote the zero-mean Gaussian process with covariance given by E [ G $_{P}$f G $_{P}$g ] = P fg − P fP g . Since Π Donsker implies Π totally bounded, Lemma A.3 guarantees the existence of a π ⋆ ∈ Π $^{⋆}$. For any s > 0 and π ⋆ s ∈ B$_{Π}$ ( π $^{⋆}$, s ) ⊂ Π, (6) then (5) combined with (A.2) yield in turn the first and second inequalities below: Since z ↦→ sup$_{f}$$_{∈F}$ | z ( f ) | is continuous and F is Donsker, the continuous mapping theo- rem [13, Theorem 1.3.6] implies that the leftmost term in the above RHS sum is O$_{P}$ ( n − 1 / $^{2}$). Therefore, (A.13) and Rem$_{n}$ = o$_{P}$ ( n − 1 / $^{2}$) imply 0 ≤ R $^{(}$̂ π$_{n}$ ) = V ⋆ − V $^{(}$̂ π$_{n}$ ) = [ V ( π ⋆ $_{s}$) − V $^{(}$̂ π$_{n}$ )] + [ V ( π $^{⋆}$) − V ( π ⋆ $_{s}$)] = ( ̂ V − V $^{)(}$̂ π$_{n}$ ) − ( ̂ V − V )( π ⋆ $_{s}$) + [ ̂ V ( π ⋆ $_{s}$) − ̂ V $^{(}$̂ π$_{n}$ )] + [ V ( π $^{⋆}$) − V ( π ⋆ $_{s}$)] ≤ ( V − V π$_{n}$ ) − ( V − V )( π ⋆ $_{s}$) + [ V ( π $^{⋆}$) − V ( π ⋆ $_{s}$)] + o$_{P}$ (Rem$_{n}$) ≲ n − 1 / 2 [ G $_{n}$f$_{̂}$ π$_{n}$ − G $_{n}$f$_{π}$ ⋆ s ] + s + [1 + o$_{P}$ (1)] Rem$_{n}$ (A.12) ≤ 2 n − 1 / $^{2}$sup f ∈F | G $_{n}$f | + s + [1 + o$_{P}$ (1)] Rem$_{n}$ . (A.13) 29 where the random terms do not depend on s . By letting s go to zero, we obtain R $^{(}$̂ π$_{n}$ ) = O$_{P}$ ( n − 1 / $^{2}$). The remainder of the proof tightens this result to R π$_{n}$ ) = o$_{P}$ ( n − 1 / $^{2}$). n $^{2}$). The remainder of the proof tightens this result to R $^{(}$̂ π$_{n}$ ) = o$_{P}$ ( n $^{2}$). Let us go back to (A.12). Since Rem$_{n}$ = o$_{P}$ ( n − 1 / $^{2}$), it also yields the tighter bound 0 ≤ R $^{(}$̂ π$_{n}$ ) ≲ s + O$_{P}$ ( n − 1 / $^{2}$) + [1 + o$_{P}$ (1)] o$_{P}$ ( n − 1 / $^{2}$) 0 ≤ R $^{(}$̂ π$_{n}$ ) ≲ n − 1 / $^{2}$lim inf s ↓ 0 inf π $^{⋆}$∈ Π ⋆ sup π ⋆ $_{s}$∈ B$_{Π}$ ( π $^{⋆}$,s ) [ G $_{n}$f$_{̂}$ π$_{n}$ − G $_{n}$f$_{π}$ ⋆ s ] + [1 + o$_{P}$ (1)] o$_{P}$ ( n − 1 / $^{2}$) . (A.14)