  Classical distribution testing assumes access to i.i.d. samples from the
distribution that is being tested. We initiate the study of Markov chain
testing, assuming access to a single trajectory of a Markov Chain. In
particular, we observe a single trajectory X0,...,Xt,... of an unknown,
symmetric, and finite state Markov Chain M. We do not control the starting
state X0, and we cannot restart the chain. Given our single trajectory, the
goal is to test whether M is identical to a model Markov Chain M0 , or far from
it under an appropriate notion of difference. We propose a measure of
difference between two Markov chains, motivated by the early work of Kazakos
[Kaz78], which captures the scaling behavior of the total variation distance
between trajectories sampled from the Markov chains as the length of these
trajectories grows. We provide efficient testers and information-theoretic
lower bounds for testing identity of symmetric Markov chains under our proposed
measure of difference, which are tight up to logarithmic factors if the hitting
times of the model chain M0 is O(n) in the size of the state space n.
