on o$_{2}$ . For P$_{0}$ almost all o$_{1}$ ∈ O , P$_{0}$ Γ TU P ( o$_{1}$, · ) is equal to where the magnitude of the big-O remainder term is upper bounded by C ‖ Rem U P ‖$_{1}$$_{,P}$$_{0}$ for a constant C > 0 which does not depend on o$_{1}$ . Taylor expansions of the first and third terms above yield where the magnitude of the big-O term can be upper bounded by C ‖ U$_{P}$ − U$_{0}$ ‖ 2 2 $_{,P$_{0}$}$. If T$_{0}$ ( O ) d = U$_{0}$ ( O ), then Recall that T,U ∈ S were arbitrary. Using that Γ$_{P}$ ≜ Γ RR P − Γ RS P − Γ SR P + Γ SS P and applying the triangle inequality gives the first result. We now turn to the second result. For any T,U ∈ S and P ∈ M , we have that where the final equality holds by a first-order Taylor expansion of ( t,u ) ↦→ e − [ t − u ] $^{2}$. The fact 52 P 2 $_{0}$Γ TU P = $^{[}$2( T$_{P}$ − U$_{P}$ ) ( U$_{0}$ − U$_{P}$ − T$_{0}$ + T$_{P}$ ) + 1 − $^{(}$4( T$_{P}$ − U$_{P}$ ) 2 − 2 $^{)}$( U$_{P}$ − U$_{0}$ )( T$_{P}$ − T$_{0}$ ) $^{]}$e − [ T$_{P}$ − U$_{P}$ ] 2 + O $^{(}$‖ L TU P ‖$_{1}$$_{,P}$$_{0}$ ) = [2( T$_{P}$ − U$_{P}$ ) ( U$_{0}$ − U$_{P}$ − T$_{0}$ + T$_{P}$ ) + 1] e − [ T$_{P}$ − U$_{P}$ ] 2 + O $^{(}$‖ L TU P ‖$_{1}$$_{,P}$$_{0}$ $^{)}$+ O $^{(}$‖ M TU P ‖ 2 2 ,P$_{0}$ ) =Φ $^{TU}$( P$_{0}$ ) + O $^{(}$‖ L TU P ‖$_{1}$$_{,P}$$_{0}$ $^{)}$+ O $^{(}$‖ M TU P ‖ 2 2 ,P$_{0}$ $^{)}$, P$_{0}$ Γ TU P ( o$_{1}$, · ) = P$_{0}$e − [ T$_{P}$ ( o$_{1}$ ) − T$_{0}$ ] 2 − 2 P$_{0}$ ( T$_{P}$ ( o$_{1}$ ) − T$_{0}$ ) D T $_{P}$( o$_{1}$ ) e − [ T$_{P}$ ( o$_{1}$ ) − T$_{0}$ ] 2 + O $^{(}$‖ Rem U P ‖$_{1}$$_{,P}$$_{0}$ $^{)}$+ O $^{(}$‖ U$_{P}$ − U$_{0}$ ‖ 2 2 ,P$_{0}$ $^{)}$. P$_{0}$ Γ TU P ( o$_{1}$, · ) = P$_{0}$e − [ T$_{P}$ ( o$_{1}$ ) − U$_{0}$ ] 2 − 2 P$_{0}$ ( T$_{P}$ ( o$_{1}$ ) − U$_{0}$ ) D T $_{P}$( o$_{1}$ ) e − [ T$_{P}$ ( o$_{1}$ ) − U$_{0}$ ] 2 + O $^{(}$‖ Rem U P ‖$_{1}$$_{,P}$$_{0}$ $^{)}$+ O $^{(}$‖ U$_{P}$ − U$_{0}$ ‖ 2 2 ,P$_{0}$ $^{)}$, P$_{0}$ [1 + 2( T$_{P}$ ( o$_{1}$ ) − U$_{P}$ )( U$_{P}$ − U$_{0}$ )] e − [ T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ] 2 + O $^{(}$‖ Rem U P ‖$_{1}$$_{,P}$$_{0}$ ) − 2 P$_{0}$ $^{[}$( T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ) + $^{(}$2( T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ) 2 − 1 $^{)}$( U$_{P}$ − U$_{0}$ ) $^{]}$D T $_{P}$( o$_{1}$ ) e − [ T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ] 2