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Suppose X, Y are random variables from probability measures F(x), G(y) respectively. The total variation distance of F and G is bounded by a constant c. Is there a way to quantify the distance between X and Y in terms of c?

The expression of this problem may not be accurate since I don't know what the answer looks like.

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    $\begingroup$ If the total variation distance between the distributions is $c$, there is a coupling of $X$ and $Y$ (that is a joint distribution for the pair $(X,Y)$ such that the first coordinate has the same marginal distribution as $X$ and the second coordinate has the same marginal distribution as $Y$) with the property that $\mathbb P(X\ne Y)=c$. This is the best possible. $\endgroup$ Jan 19, 2017 at 0:12
  • $\begingroup$ Thanks Anthony! Your coupling idea seems to be the exact solution to the problem. By "best possible" do you mean that there is no other coupling pair to make the probability of "unequal" smaller than c? Is there a formal name of this result? $\endgroup$ Jan 20, 2017 at 2:37
  • $\begingroup$ The result is Lemma 4.9 in Sebastian Roch's lecture notes, where he calls it the coupling inequality. math.wisc.edu/~roch/mdp/index.html $\endgroup$ Jan 20, 2017 at 7:47

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