Let (X_n) be a Markov chain and let L_n denote its empirical measure at time n. We are interested in the large deviations of (L_n). Roughly speaking, proving a large deviation principle for (L_n) means proving that, for any given measure \mu, the probability of L_n being close to \mu decays with n at an exponential rate (depending on \mu). The large deviations of (L_n) have been studied since the 1970s and are well understood in "good" cases, in particular under assumptions of irreducibility of the Markov chain. However, very some simple Markov chains fail to satisfy these irreducibility assumptions. It turns out that transient states may play a role in large deviations, and complex behaviours can emerge at the large deviations scale when the Markov chain is not irreducible. I will describe these behaviours and present a new method for deriving the weak large deviation principle for (L_n) in the reducible case, despite the resulting complication.
