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We study the problem of hypothesis testing between two discretedistributions, where we only have access to samples after the action of a knownreversible Markov chain, playing the role of noise. We deriveinstance-dependent minimax rates for the sample complexity of this problem, andshow how its dependence in time is related to the spectral properties of theMarkov chain. We show that there exists a wide statistical window, in terms ofsample complexity for hypothesis testing between different pairs of initialdistributions. We illustrate these results in several concrete examples.

Statistical Windows in Testing for the Initial Distribution of a Reversible Markov Chain