also expect (S3) to hold if P ↦→ ∫ T$_{P}$ ( X $^{T}$( o )) dP ( o ) is pathwise differentiable with canonical gradient uniformly bounded in P and o in the model in which the marginal distribution of X is known. The canonical gradient in this model is equal to D T $_{P}$. As indicated above, if σ$_{0}$ = 0, the behavior of Ψ around P$_{0}$ cannot be adequately character- ized by a first-order analysis. For this reason, we must investigate whether Ψ is second-order differentiable. As we discuss below, under H$_{0}$ , Ψ is indeed second-order pathwise differen- tiable at P$_{0}$ and admits a useful second-order asymptotic representation. Theorem 2 (Second-order pathwise differentiability under H$_{0}$ ) . If R, S ∈ S and H$_{0}$ holds, the parameter Ψ : M → R is second-order pathwise differentiable at P$_{0}$ with second-order canonical gradient D Ψ 2 ( P$_{0}$ ) ≜ 2Γ$_{0}$ . It is easy to confirm that Γ$_{0}$, and thus D Ψ 2 , is one-degenerate under H$_{0}$ in the sense that ∫ Γ$_{0}$( o, o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) = ∫ Γ$_{0}$( o$_{1}$, o ) dP$_{0}$ ( o$_{1}$ ) = 0 for all o . This is shown as follows. For any T, U ∈ S , the law of total expectation conditional on X U and fact that ∫ D U 0 ( o ) dP$_{0}$ ( o | x $^{U}$) = 0 yields that where we have written Γ TU 0 to denote Γ TU P$_{0}$ . Since ∫ f ( R$_{0}$ ( o )) dP$_{0}$ ( o ) = ∫ f ( S$_{0}$ ( o )) dP$_{0}$ ( o ) for each measurable function f when S$_{0}$ ( O ) d = T$_{0}$ ( O ), this then implies that ∫ Γ RS 0 ( o, o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) = ∫ Γ RR 0 ( o, o$_{2}$ ) dP ( o$_{2}$ ) and ∫ Γ SR 0 ( o, o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) = ∫ Γ SS 0 ( o, o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) under H$_{0}$ . Hence, it fol- lows that ∫ Γ$_{0}$( o, o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) = 0 under H$_{0}$ for any o . If second-order pathwise differentiability held in a sufficiently uniform sense over M , we 11 ∫ Γ TU 0 ( o, o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) = ∫ $^{{}$1 − 2 [ T$_{0}$ ( o ) − U$_{0}$ ( o$_{2}$ )] D T 0 ( o ) $^{}}$e − [ T$_{0}$ ( o ) − U$_{0}$ ( o$_{2}$ )] $^{2}$dP$_{0}$ ( o$_{2}$ ) , 2.3 Second-order differentiability and asymptotic representation