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 The first two terms in the last equality are equal to We now look to find the portion of the canonical gradient given by the third term. We have that 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$_{0}$ . d dt Φ $^{TU}$( P$_{t}$ ) ∣ ∣ t =0 = ∫ [ ∫ e − [ T$_{0}$ ( x T 1 ) − U$_{0}$ ( x U 2 )] $^{2}$dP$_{0}$ ( x T $_{1}$)$^{]}$ h$_{1}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) + ∫ [ ∫ e − [ T$_{0}$ ( x T 1 ) − U$_{0}$ ( x U 2 )] $^{2}$dP$_{0}$ ( x U 2 )$^{]}$ h$_{1}$ ( o$_{1}$ ) dP$_{0}$ ( o$_{1}$ ) − 2$^{∫}$ ∫ ( T$_{0}$ ( x T $_{1}$) − U$_{0}$ ( x U 2 ))( ˙ T$_{0}$ ( x T $_{1}$) − ˙ U$_{0}$ ( x U 2 )) e − [ T$_{0}$ ( x T 1 ) − U$_{0}$ ( x U 2 )] $^{2}$dP$_{0}$ ( x U 2 ) dP$_{0}$ ( x T $_{1}$) . First term = ∫ $^{(}$E$_{P}$$_{0}$ $^{[}$e − [ T$_{0}$ ( X $^{T}$) − U$_{0}$ ( x $^{U}$)] 2 ] − E$_{P}$ 2 0 $^{[}$e − [ T$_{0}$ ( X T 1 ) − U$_{0}$ ( X U 2 )] 2 $^{])}$h$_{1}$ ( o ) dP$_{0}$ ( o ) Second term = ∫ $^{(}$E$_{P}$$_{0}$ $^{[}$e − [ T$_{0}$ ( x $^{T}$) − U$_{0}$ ( X $^{U}$)] 2 ] − E$_{P}$ 2 0 $^{[}$e − [ T$_{0}$ ( X T 1 ) − U$_{0}$ ( X U 2 )] 2 $^{])}$h$_{1}$ ( o ) dP$_{0}$ ( o ) . − 2$^{∫}$ ∫ ( T$_{0}$ ( x T $_{1}$) − U$_{0}$ ( x U 2 )) ˙ T$_{0}$ ( x T $_{1}$) e − [ T$_{0}$ ( x T 1 ) − U$_{0}$ ( x U 2 )] $^{2}$dP$_{0}$ ( x U 2 ) dP$_{0}$ ( x T $_{1}$) = ∫ 2 E$_{P}$$_{0}$ $^{[}$( U$_{0}$ ( X $^{U}$) − T$_{0}$ ( x $^{T}$)) e − [ T$_{0}$ ( x $^{T}$) − U$_{0}$ ( X $^{U}$)] 2 $^{]}$D T 0 ( o ) h$_{1}$ ( o ) dP$_{0}$ ( o ) 2$^{∫}$ ∫ ( T$_{0}$ ( x T $_{1}$) − U$_{0}$ ( x U 2 )) ˙ U$_{0}$ ( x U 2 ) e − [ T$_{0}$ ( x T 1 ) − U$_{0}$ ( x U 2 )] $^{2}$dP$_{0}$ ( x U 2 ) dP$_{0}$ ( x T $_{1}$) = ∫ 2 E$_{P}$$_{0}$ $^{[}$( T$_{0}$ ( X $^{T}$) − U$_{0}$ ( x $^{U}$)) e − [ T$_{0}$ ( X $^{T}$) − U$_{0}$ ( x $^{U}$)] 2 $^{]}$D U 0 ( o ) h$_{1}$ ( o ) dP$_{0}$ ( o ) . Φ $^{TU}$( P$_{t}$ ) − Φ $^{TU}$( P$_{0}$ ) = tE$_{P}$$_{0}$ $^{[}$D $^{TU}$( O ) h$_{1}$ ( O ) ] + o ( t ) . 42