content_image/324055.jpg content_image/324054.jpg content_image/324053.jpg content_image/324052.jpg
The final inequality above holds by the FKG inequality (Fortuin et al., 1971). It follows that 

<img src='content_image/324055.jpg'>

where the final equality holds under H$_{0}$ by a Taylor expansion of z ↦→ z (2 z 2 − 3) e − z 2 and analogous calculations to those used in (A.10). We note that the second equality above uses a different F and a different big-O term than the line above, and that the big-O term can be upper bounded by C ‖ M TU P ‖ 4 4 ,P$_{0}$ for a constant C > 0. 

Plugging (A.8), (A.9), and (A.11) into (A.7), applying the triangle inequality, and using the bounds on B gives the result. 

We give a lemma before proving Theorem 3. 

Lemma A.6. Let K$_{P}$ ≜ ‖ L RS P ‖$_{1}$$_{,P}$$_{0}$ + ‖ M RS P ‖ 2 2 ,P$_{0}$ for all P ∈ M . If H$_{0}$ holds, then for all P ∈ M , 

<img src='content_image/324054.jpg'>

where O ′ ⊆ O is some P$_{0}$ probability 1 set. More generally, for all P$_{0}$ ∈ M , 

Proof of A.6. For T,U ∈ S , we have that 

<img src='content_image/324053.jpg'>

<img src='content_image/324052.jpg'>

Above we have omitted the dependence of Γ TU on ( o$_{1}$,o$_{2}$ ), T and D T P on o$_{1}$ , and U and D U P 

