B ( o$_{1}$,o$_{2}$ ) and B$_{0}$ ( o$_{1}$,o$_{2}$ ) for all o$_{1}$,o$_{2}$ such that where the second equality holds under H$_{0}$ . The boundedness of ˙ f$_{1}$ in [ − 2 , 2], the triangle inequality, and the Cauchy-Schwarz inequality yield A Taylor expansion of f$_{2}$ ( z ) = (2 z 2 − 1) e − z 2 yields that there exists a ˜ B$_{2}$ that falls between B and B$_{0}$ such that The first line on the right is equal to F ( P,P$_{0}$,T,U ) under H$_{0}$ . By the triangle inequality and the boundedness of ¨ f$_{2}$ on [ − 2 , 2], the third line satisfies 50 E$_{P}$ 2 0 [ $^{(}$[ T − T$_{0}$ ] 2 + [ U − U$_{0}$ ] 2 $^{)}$( B − B$_{0}$ ) 2 ¨ f$_{2}$ ( ˜ B$_{2}$ ) 2 ] ≲ 4 ∑ k =0 E$_{P}$ 2 0 ∣ $_{∣}$[ T − T$_{0}$ ] k [ U − U$_{0}$ ] 4 − k ∣ ∣ ≲ 4 ∑ k =0 E$_{P}$$_{0}$ ∣ $_{∣}$[ M TU P ] k ∣ $_{∣}$E$_{P}$$_{0}$ ∣ $_{∣}$[ M TU P ] 4 − k ∣ ≲ ‖ M TU P ‖ 4 4 $_{,P$_{0}$}$. (A.10) E$_{P}$ 2 0 [ $^{(}$[ T − T$_{0}$ ] 2 + [ U − U$_{0}$ ] 2 $^{) (}$2 B 2 − 1 $^{)}$e − B 2 ] = E$_{P}$ 2 0 [ $^{(}$[ T − T$_{0}$ ] 2 + [ U − U$_{0}$ ] 2 $^{) (}$2 B 2 0 − 1 $^{)}$e − B 2 $_{0}$] + 2 E$_{P}$ 2 0 [ $^{(}$[ T − T$_{0}$ ] 2 + [ U − U$_{0}$ ] 2 $^{)}$( B − B$_{0}$ ) $^{(}$B (2 B 2 − 3) $^{)}$e − B 2 ] + E$_{P}$ 2 0 [ $^{(}$[ T − T$_{0}$ ] 2 + [ U − U$_{0}$ ] 2 $^{)}$( B − B$_{0}$ ) 2 ¨ f$_{2}$ ( ˜ B$_{2}$ ) 2 ] . E$_{P}$ 2 0 ∣ ∣ $^{(}$Rem U P − Rem T P $^{)}$( B − B$_{0}$ ) ˙ f$_{1}$ ( ˜ B ) ∣ ∣ ≲ E$_{P}$ 2 0 ∣ $^{(}$Rem U P − Rem T P $^{)}$( B − B$_{0}$ ) ∣ ≲ E$_{P}$ 2 0 ∣ $_{∣}$L TU P ( O$_{1}$ ) M TU P ( O$_{2}$ ) ∣ + E$_{P}$$_{0}$ ∣ $_{∣}$L TU P ∣ $_{∣}$E$_{P}$$_{0}$ ∣ $_{∣}$M TU P ∣ ≲ ‖ L TU P ‖$_{2}$$_{,P}$$_{0}$ ‖ M TU P ‖$_{2}$$_{,P}$$_{0}$ . (A.9) E$_{P}$ 2 0 $^{[}$2 B $^{(}$Rem U P − Rem T P $^{)}$e − B 2 ] = E$_{P}$ 2 0 [ $^{(}$Rem U P − Rem T P ) $^{(}$2 B$_{0}$e − B 2 0 + ( B − B$_{0}$ ) ˙ f$_{1}$ ( ˜ B ) )] = F ( P,P$_{0}$,T,U ) + E$_{P}$ 2 0 [ $^{(}$Rem U P − Rem T P $^{)}$( B − B$_{0}$ ) ˙ f$_{1}$ ( ˜ B ) $^{]}$, (A.8)