The above remainder is double robust in the sense that it is zero if either the treatment mechanism (i.e., the probability of A given W ) or the outcome regression (i.e., the expected value of Y given A and W ) is correctly specified under P . In a randomized trial where the treatment mechanism is known and specified correctly in P , we have that Rem R P ≡ 0 and thus L RS P ≡ 0. More generally, an upper bound for Rem R P can be found using the Cauchy- Schwarz inequality to relate the rate of ‖ Rem R P ‖$_{2}$$_{,P}$$_{0}$ to the product of the L $^{2}$( P$_{0}$ )-norm for the difference between each of the treatment mechanism and the outcome regression under P and P$_{0}$ . For (S1) we take x R = x S = w . Condition (S2) can be verified using an expression similar to that in (8). Condition (S3) is satisfied by If min$_{a}$ P ( A = a | W ) is bounded away from zero with probability 1 uniformly in P , both ( P,o ) ↦→ D R $_{P}$( o ) and ( P,o ) ↦→ D S $_{P}$( o ) are uniformly bounded. Similarly to Example 1, we have that Rem R $_{P}$( o ) is equal to The remainder Rem S $_{P}$( o ) is equal to the above display but with A = 1 replaced by A = 0. The discussion about the double robust remainder term from Example 1 applies to these remainders as well. 24 Example 3 (Continued). Example 2 (Continued). D R $_{P}$( o ) ≜ a P ( A = a | W = w ) [ y − E$_{P}$ ( Y | A = a,W = w )] D S $_{P}$( o ) ≜ 1 − a P ( A = a | W = w ) [ y − E$_{P}$ ( Y | A = a,W = w )] . E$_{P}$$_{0}$$^{{[}$ 1 − P$_{0}$ ( A = 1 | W ) P ( A = 1 | W ) ] [ E$_{P}$ ( Y | A,W ) − E$_{P}$$_{0}$ ( Y | A,W )] ∣ ∣ A = 1 ,W = w$^{}}$.