Accordingly, let where ˜ V$_{n}$ is an estimator of V 0 . Then where, by construction, ˜ G ∗ $_{n}$( s ) = G ∗ $_{n}$( s ) and ˜ Q ∗ $_{n}$( s ) = − s ′ ˜ V$_{n}$s / 2 . Because ˜ G ∗ n = G ∗ $_{n}$, ˜ G ∗ $_{n}$( s ) ⇝ P G ( s ) whenever (7) holds. In addition, ˜ Q ∗ $_{n}$( s ) → P Q ( s ) provided ˜ V$_{n}$ → P V 0 . As a consequence, it seems plausible that the distributional approximation is consistent if ˜ V$_{n}$ → P V 0 . To make the heuristics of the previous section precise, we impose the following condition. 3 Main Result ˜ m$_{n}$ ( z , θ ) = m ( z , θ ) − M$_{n}$ ( θ ) − 1 2 ( θ − ˆ θ$_{n}$ ) ′ ˜ V$_{n}$ ( θ − ˆ θ$_{n}$ ) , 3 √$_{n}$$_{(}$ ˜ θ ∗ n − ˆ θ$_{n}$ ) = argmax s ∈ R d { ˜ Q ∗ $_{n}$( s ) + ˜ G ∗ $_{n}$( s ) } , P [ 3 √$_{n}$$_{(}$ ˆ θ$_{n}$ − θ 0 ) ≤ t ] ≈ P $^{∗}$[ 3 √$_{n}$$_{(}$ ˜ θ ∗ n − ˆ θ$_{n}$ ) ≤ t ] H ( s , t ) = lim θ → θ 0 ,δ ↓ 0 1 δ E [ { m ( z , θ + δ s ) − m ( z , θ ) }{ m ( z , θ + δ t ) − m ( z , θ ) } ] exists, and H ( s , s ) + H ( t , t ) − 2 H ( s , t ) > 0 for all s = t . (vi) For R near zero, the classes D R = { m ( · , θ ) − m ( · , θ 0 ) : ‖ θ − θ 0 ‖ ≤ R } are uniformly 6 Also, the class M = { m ( · , θ ) : θ ∈ Θ } is manageable for the envelope ¯ M ( · ) = sup$_{m}$$_{∈M}$ | m ( · ) | , and E [ ¯ M ( z ) $^{4}$] < ∞ . Condition CRA (Cube Root Asymptotics) (i) For every δ > 0 , sup$_{‖}$$_{θ}$$_{−}$$_{θ}$$_{0}$$_{‖}$$_{>δ}$ M ( θ ) < M ( θ 0 ) . (ii) θ 0 is an interior point of Θ . (iii) M is twice continuously differentiable near θ 0 , with V 0 positive definite. (iv) For θ 1 , θ 2 near θ 0 , E [ | m ( z , θ 1 ) − m ( z , θ 2 ) | ] = O ( ‖ θ 1 − θ 2 ‖ ) . (v) For all s , t ∈ R $^{d}$,