content_image/319828.jpg content_image/319827.jpg content_image/319826.jpg
## Proofs for Section 2 

We give two lemmas before proving Theorem 1. 

Lemma A.1. For any T, U ∈ S and any fluctuation submodel dP$_{t}$ = (1 + th$_{1}$ + t $^{2}$h$_{2}$ ) dP$_{0}$ , we have that, for all ˜ t in a neighborhood of zero, 

<img src='content_image/319828.jpg'>

Proof of A.1. We have that 

<img src='content_image/319827.jpg'>

where the derivative is passed under the integral in view of (S2). The result follows by the chain rule. 

For each T, U ∈ S , define 

<img src='content_image/319826.jpg'>

We have omitted the dependence of D TU on P$_{0}$ in the notation. We first give a key lemma about the parameter Φ $^{TU}$. 

Lemma A.2 (First-order canonical gradient of Φ $^{TU}$) . Let T and U be members of S . Then Φ TU has canonical gradient D TU at P$_{0}$ . 

