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D Ψ 2 ( o$_{2}$, o$_{1}$ ) for all o$_{1}$, o$_{2}$ . The arguments given below the theorem statement in the main text establish the one-degeneracy of Γ$_{0}$ under H$_{0}$ show that E$_{P}$$_{0}$ [ D Ψ 2 ( O, o )] = E$_{P}$$_{0}$ [ D Ψ 2 ( o, O )] = 0 for all o ∈ O under H$_{0}$ . Condition (S2) ensures that ‖ D Ψ 2 ‖$_{2}$$_{,P}$ 2 0 < ∞ , and thus D Ψ 2 is P 2 0 square integrable and one-degenerate. 

Because the first pathwise derivative is zero under the null, we have that 

Thus D Ψ 2 is a second-order canonical gradient of Ψ at P$_{0}$ . 

We give a lemma before proving Theorem 3. 

Lemma A.5. Fix P ∈ M . For all T, U ∈ S , let 

<img src='content_image/297029.jpg'>

<img src='content_image/297030.jpg'>

There exists a mapping ζ ( P, P$_{0}$, · ) : S →$_{R}$ such that, for all T, U ∈ S for which T$_{0}$ ( O ) d = U$_{0}$ ( O ) , 

<img src='content_image/297031.jpg'>

Proof of A.5. In this proof we use F ( P, P$_{0}$, T, U ) to denote any constant which can be writ- ten as ˜ ζ ( P, P$_{0}$, T ) + ˜ ζ ( P, P$_{0}$, U ) for expressions ˜ ζ ( P, P$_{0}$, T ) and ˜ ζ ( P, P$_{0}$, U ) which satisfy ˜ ζ ( P, P$_{0}$, T ) = ˜ ζ ( P, P$_{0}$, U ) whenever T = U . We will write c$_{1}$F ( P, P$_{0}$, T, U )+ c$_{2}$F ( P, P$_{0}$, T, U ) = F ( P, P$_{0}$, T, U ) for any real numbers c$_{1}$, c$_{2}$ . We then fix ζ to be the final instance of ˜ ζ upon exiting the proof. 

Fix T, U ∈ S . Let b$_{0}$ ( o$_{1}$, o$_{2}$ ) ≜ T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ ) and b ( o$_{1}$, o$_{2}$ ) ≜ T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ( o$_{2}$ ) for any o$_{1}$, o$_{2}$ . For ease of notation, in the expected values below we will write B and B$_{0}$ to refer to b ( O$_{1}$, O$_{2}$ ) and b$_{0}$ ( O$_{1}$, O$_{2}$ ), respectively. We also write T for T$_{P}$ ( O$_{1}$ ), T$_{0}$ for T$_{0}$ ( O$_{1}$ ), Rem T P for Rem T $_{P}$( O$_{1}$ ), U for U$_{P}$ ( O$_{2}$ ), U$_{0}$ for U$_{0}$ ( O$_{2}$ ), and Rem U P for Rem U $_{P}$( O$_{2}$ ). 

