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We now show the necessity of the stated conditions. Suppose that σ$_{0}$ = 0 and H$_{0}$ does not hold. It is easy to verify that 

<img src='content_image/310185.jpg'>

is a first-order gradient in the model where R$_{0}$ and S$_{0}$ are known (possibly an inefficient gradient depending on the form of R and S ). Call the variance of this gradient ˜ σ$_{0}$ . As the model where R$_{0}$ and S$_{0}$ are known is a submodel of the (locally) nonparametric model, ˜ σ$_{0}$ ≤ σ$_{0}$ , and hence ˜ σ$_{0}$ = 0 and ˜ D Ψ 1 ≡ 0. Now, if ˜ σ$_{0}$ = 0 and H$_{0}$ does not hold, then A.3 shows that R$_{0}$ ( O ) and S$_{0}$ ( O ) are degenerate. Finally, ˜ D Ψ 1 ≡ 0 and the degeneracy of R$_{0}$ ( O ) and S$_{0}$ ( O ) shows that for almost all o , 

<img src='content_image/310184.jpg'>

where we use r$_{0}$ and s$_{0}$ to denote the (probability 1) values of R$_{0}$ ( O ) and S$_{0}$ ( O ). The above is zero almost surely if and only if D R 0 ≡ D S $_{0}$. Thus σ$_{0}$ = 0 only if (i) or (ii) holds. 

We give the following lemma before proving Theorem 2. Before giving the lemma, we define the function Π : S →$_{R}$ . Suppressing the dependence on P$_{0}$ and h$_{1}$ , h$_{2}$ , for all V ∈ S and t ̸ = 0 we define 

<img src='content_image/310183.jpg'>

Lemma A.4. For any fluctuation submodel consistent with (A.4), T, U ∈ S with T$_{0}$ ( O ) d = U$_{0}$ ( O ) , 

