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Accordingly, let 

where ˜ V$_{n}$ is an estimator of V 0 . Then 

<img src='content_image/315957.jpg'>

<img src='content_image/315958.jpg'>

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 . 

## 3 Main Result 

<img src='content_image/315959.jpg'>

To make the heuristics of the previous section precise, we impose the following condition. 

Condition CRA (Cube Root Asymptotics) (i) For every δ > 0 , sup$_{‖}$$_{θ}$$_{−}$$_{θ}$$_{0}$$_{‖}$$_{>δ}$ M ( θ ) < M ( θ 0 ) . 

Also, the class M = { m ( · , θ ) : θ ∈ Θ } is manageable for the envelope ¯ M ( · ) = sup$_{m}$$_{∈M}$ | m ( · ) | , and E [ ¯ M ( z ) $^{4}$] < ∞ . 

(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 $^{d}$, 

exists, and H ( s , s ) + H ( t , t ) − 2 H ( s , t ) > 0 for all s ̸ = t . 

<img src='content_image/315960.jpg'>

(vi) For R near zero, the classes D R = { m ( · , θ ) − m ( · , θ 0 ) : ‖ θ − θ 0 ‖ ≤ R } are uniformly 

