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Sep 23, 2016 at 17:39 comment added Yuval Peres This is essentially correct, but there might be some confusion between the product measures $p^n$ and $q^n$, and the multinomial measures that are their projections. The reason the proof works is that a symmetry argument shows that the total variation distance is not changed by the projection. Working directly with the multinomial measures would require a proof of the Martingale property since the corresponding $\sigma$ fields are not increasing.
Sep 20, 2016 at 13:52 comment added Alainty Ok, right! Thank you very much michael!!
Sep 20, 2016 at 7:01 history answered user83457 CC BY-SA 3.0