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Can someone help me with the following problem:

Let $P_n$ and $Q_n$ two multinomial laws with parameters $(p,n)$ and $(q,n)$, where $p$ and $q$ are two probability measures on some measurable space and $n\in \mathbb{N}$. Is it true that $\|P_n-Q_n\|_{TV}$ is non decreasing in $n$? I think that it is, but I cannot prove it...

Thank you.

Alainty

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  • $\begingroup$ @SergueiPopov, Alainty it seems I did misunderstand the question so i deleted my answer. $\endgroup$
    – usul
    Sep 19, 2016 at 23:32

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Assume both measures put positive probability on all outcomes. The total variation distance can be written $E_Q|1-\frac {dP_n}{dQ_n}|$. The likelihood ratio is a martingale, so the integrand is a submartingale and so it's expectation is increasing.

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  • $\begingroup$ Ok, right! Thank you very much michael!! $\endgroup$
    – Alainty
    Sep 20, 2016 at 13:52
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    $\begingroup$ 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. $\endgroup$
    – Yuval Peres
    Sep 23, 2016 at 17:39

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