1 t q ETV2(ˆQt, Q): =C<+ entails T: = 1 T T1X t=1 q ETV2(ˆQt, Q) 0. Indeed, 1 T T1X t=1 q ETV2(ˆQt, Q) = 1 T

This supplementary material contains all the proofs omitted from the main body. Furthermore, the Cauchy-Schwarz inequality, followed by an application of the bound of Eq.(8), Wesubstitute the bound from Eq.(18),keeping in mind that the total variation distance is always smallerthan1: E[ST]6E d2T + K T We observe that Eq.(20) trivially holds fort = t . Assume that Eq.(20) holds fort > t . We takeS = A = B = {0,1}and letX be an arbitrary finite set. As px(1) [0,1], the fact that the first integral above is larger than1 entails thatpx(1) = 1 on supp(Q0).