We study the problem of hypothesis testing between two discrete
distributions, where we only have access to samples after the action of a known
reversible Markov chain, playing the role of noise. We derive
instance-dependent minimax rates for the sample complexity of this problem, and
show how its dependence in time is related to the spectral properties of the
We show that there exists a wide statistical window, in terms of
sample complexity for hypothesis testing between different pairs of initial
distributions. We illustrate these results in several concrete examples.