The expression inside the second pair of integrals only depends on o$_{1}$ through T ( o$_{1}$ ). Thus we can rewrite this term as E$_{P}$$_{0}$ [ f ( T ( O$_{1}$ ))] for a fixed function f that relies on P$_{0}$ , h$_{1}$ , h$_{2}$ , and U . Under H$_{0}$ , we can rewrite this term as E$_{P}$$_{0}$ [ f ( U ( O$_{1}$ ))]. That is, we can replace each T ( O$_{1}$ ) in the second pair of integrals with U ( O$_{1}$ ). This yields Π( U ). Switching the roles of o$_{1}$ and o$_{2}$ in the first pair of integrals above and applying Fubini’s theorem shows that By the same arguments used to for the second pair of integrals, the above expression is equal to Π( T ) under H$_{0}$ . By (S3), the third pair of integrals can be rewritten as Proof of Theorem 2. We start by noting that 1 2 d 2 dt $_{2}$ψ$_{t}$ ∣ ∣ t =0 is equal to where the penultimate equality makes use of A.4. It is easy to verify that D Ψ 2 ( o$_{1}$, o$_{2}$ ) = 47 1 2 [ d 2 dt $_{2}$Φ $^{TT}$( P$_{t}$ )$^{∣}$ ∣ ∣ t =0 + d 2 dt $_{2}$Φ $^{UU}$( P$_{t}$ )$^{∣}$ ∣ ∣ t =0 − d 2 dt $_{2}$Φ $^{TU}$( P$_{t}$ )$^{∣}$ ∣ ∣ t =0 − d 2 dt $_{2}$Φ $^{UT}$( P$_{t}$ )$^{∣}$ ∣ ∣ t =0 ] = ∫ ∫ $^{[}$Γ RR 0 ( o$_{1}$, o$_{2}$ ) + Γ SS 0 ( o$_{1}$, o$_{2}$ ) − Γ RS 0 ( o$_{1}$, o$_{2}$ ) − Γ SR 0 ( o$_{1}$, o$_{2}$ ) $^{]}$h$_{1}$ ( o$_{1}$ ) h$_{1}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) = 1 2 ∫ ∫ D Ψ 2 ( o$_{1}$, o$_{2}$ ) h$_{1}$ ( o$_{1}$ ) h$_{1}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) , 2$^{∫}$ ∫ $^{[}$2( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) $^{(}$D U 0 ( o$_{2}$ ) − D T 0 ( o$_{1}$ ) ) − $^{(}$4( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) 2 − 2 $^{)}$D T 0 ( o$_{1}$ ) D U 0 ( o$_{2}$ ) + 1 ] × e − [ T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )] $^{2}$h$_{1}$ ( o$_{1}$ ) h$_{1}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) . 2$^{∫}$ ∫ $^{[}$2( T$_{0}$ ( o$_{2}$ ) − U$_{0}$ ( o$_{1}$ )) ˙ T$_{0}$ ( o$_{2}$ ) h$_{1}$ ( o$_{2}$ ) + 2( T$_{0}$ ( o$_{2}$ ) − U$_{0}$ ( o$_{1}$ )) 2 ˙ T$_{0}$ ( o$_{2}$ ) 2 + h$_{2}$ ( o$_{2}$ ) − ˙ T$_{0}$ ( o$_{2}$ ) 2 + ( T$_{0}$ ( o$_{2}$ ) − U$_{0}$ ( o$_{1}$ )) ¨ T$_{0}$ ( o$_{2}$ ) $^{]}$e − [ T$_{0}$ ( o$_{2}$ ) − U$_{0}$ ( o$_{1}$ )] $^{2}$dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) . + 2$^{∫}$ ∫ $^{[}$2( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) ( ˙ U$_{0}$ ( o$_{2}$ ) h$_{1}$ ( o$_{1}$ ) − ˙ T$_{0}$ ( o$_{1}$ ) h$_{1}$ ( o$_{2}$ ) − $^{(}$4( T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )) 2 − 2 ) ˙ T$_{0}$ ( o$_{1}$ ) ˙ U$_{0}$ ( o$_{2}$ ) + h$_{1}$ ( o$_{1}$ ) h$_{1}$ ( o$_{2}$ ) $^{]}$e − [ T$_{0}$ ( o$_{1}$ ) − U$_{0}$ ( o$_{2}$ )] $^{2}$dP$_{0}$ ( o$_{2}$ ) dP$_{0}$ ( o$_{1}$ ) .