Total variation revisited

This is a follow-up on my earlier post on the total variation distance. As I already mentioned, my visit to Stanford and Berkeley were immensely useful, not least because of an opportunity to meet with the experts in a field to which I aspire to contribute. In that earlier post on total variation, I gave some characterizations and properties of TV, fully aware of the low likelihood that these are original observations. Sure enough, the relation

has been known for quite some time; see the book-in-progress by Aldous and Fill, or the one by Pollard.

The relation

also seems to be folklore knowledge; I have not seen a proof anywhere and give a simple (non-probabilistic) one here (Lemma 2.6). Amir Dembo suggested that I re-derive this in a probability-theoretic way, via coupling. Here it is.

Recall that if and are probability measures on then
,
where the infimum is taken over all the distributions on , having marginals and , resp., and the random variables are and are distributed and . Any such joint measure on is called a coupling and one achieving the infimum is called a maximal coupling.

Applying this to our situation, let us define the random variables . Let be a maximal coupling of and , and define similarly for and . Notice that is a (not necessarily maximal) coupling of and . Then

.