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Proof of A.2. To consider first-order behavior it suffices to consider fluctuation submodels in which h$_{2}$ ( o ) = 0 for all o . We first derive the first-order pathwise derivative of the parameter Φ TU at P$_{0}$ . Applying the preceding lemma at ˜ t = 0 yields that 

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The first two terms in the last equality are equal to 

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We now look to find the portion of the canonical gradient given by the third term. We have that 

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Collecting terms, a first-order Taylor expansion of t ↦→ Φ $^{TU}$( P$_{t}$ ) about t = 0 yields that 

Thus Φ TU has canonical gradient D TU at P$_{0}$ . 

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