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- Mar 10, 2012

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Assume that there is an $\varepsilon>0$ such that whenever $\mu_0$ and $\nu_o$ are point distributions on $S$ (in other words, $\mu_0$ and $\nu_0$ are Direac masses) we have

$$\|\mu_0P-\nu_0P\|_{TV}\leq \varepsilon$$

Now let $Y=Y_0, Y_1, Y_2, \ldots$ be an independent copy of $X$.

**Question.**What can we say about the magnitude of $P[X_1\neq Y_1|X_0=x_0, Y_0=y_0]$, where $x_0$ and $y_0$ are two different states in $S$.

I intuitively think that we should be above to bound the above quantity by $\varepsilon$ up to multiplication by an absolute constant. But I am unable to prove it.

We have that

$$P[X_1\neq Y_1|X_0=x_0, Y_0=y_0] = \sum_{x\in S}\sum_{y\in S, y\neq x}P[X_1=x, Y_1=y|X_0=x_0, Y_0=y_0]$$

which is equal to

$$\sum_{x\in S} p(x_0, x)(1-p(y_0, x))$$

but I am unable to make progress.