content_image/301212.jpg content_image/301211.jpg content_image/301210.jpg content_image/301209.jpg
on o$_{2}$ . For P$_{0}$ almost all o$_{1}$ ∈ O , P$_{0}$ Γ TU P ( o$_{1}$, · ) is equal to 

<img src='content_image/301212.jpg'>

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 

<img src='content_image/301211.jpg'>

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 

<img src='content_image/301210.jpg'>

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 

<img src='content_image/301209.jpg'>

where the final equality holds by a first-order Taylor expansion of ( t,u ) ↦→ e − [ t − u ] $^{2}$. The fact 

