  Given samples from two distributions over an $n$-element set, we wish to test
whether these distributions are statistically close. We present an algorithm
which uses sublinear in $n$, specifically, $O(n^{2/3}\epsilon^{-8/3}\log n)$,
independent samples from each distribution, runs in time linear in the sample
size, makes no assumptions about the structure of the distributions, and
distinguishes the cases when the distance between the distributions is small
(less than $\max\{\epsilon^{4/3}n^{-1/3}/32, \epsilon n^{-1/2}/4\}$) or large
(more than $\epsilon$) in $\ell_1$ distance. This result can be compared to the
lower bound of $\Omega(n^{2/3}\epsilon^{-2/3})$ for this problem given by
Valiant.
  Our algorithm has applications to the problem of testing whether a given
Markov process is rapidly mixing. We present sublinear for several variants of
this problem as well.
