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 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 , 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 Lemma A.4. For any fluctuation submodel consistent with (A.4), T, U ∈ S with T$_{0}$ ( O ) d = U$_{0}$ ( O ) , Π( V ) ≜ 2$^{∫}$ ∫ $^{[}$2( V$_{0}$ ( o$_{2}$ ) − V$_{0}$ ( o$_{1}$ )) ˙ V$_{0}$ ( o$_{2}$ ) h$_{1}$ ( o$_{2}$ ) + 2( V$_{0}$ ( o$_{2}$ ) − V$_{0}$ ( o$_{1}$ )) 2 ˙ V$_{0}$ ( o$_{2}$ ) 2 + h$_{2}$ ( o$_{2}$ ) − ˙ V$_{0}$ ( o$_{2}$ ) 2 + ( V$_{0}$ ( o$_{2}$ ) − V$_{0}$ ( o$_{1}$ )) ¨ V$_{0}$ ( o$_{2}$ ) $^{]}$e − [ V$_{0}$ ( o$_{2}$ ) − V$_{0}$ ( o$_{1}$ )] $^{2}$dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) . D Ψ 1 ( o ) = 2 D $^{RS}$( o ) = 2( s$_{0}$ − r$_{0}$ ) $^{(}$D R 0 ( o ) − D S $_{0}$( o ) $^{)}$e − [ r$_{0}$ − s$_{0}$ ] $^{2}$, ˜ D Ψ 1 ≜ E$_{P}$$_{0}$ $^{[}$e − [ R$_{0}$ ( O ) − R$_{0}$ ( o )] 2 $^{]}$+ E$_{P}$$_{0}$ $^{[}$e − [ S$_{0}$ ( O ) − S$_{0}$ ( o )] 2 ] − E$_{P}$$_{0}$ $^{[}$e − [ R$_{0}$ ( O ) − S$_{0}$ ( o )] 2 ] − E$_{P}$$_{0}$ $^{[}$e − [ R$_{0}$ ( o ) − S$_{0}$ ( O )] 2 ] − ψ$_{0}$ 45