LIMIT THEOREMS FOR MARKOV WALKS CONDITIONED TO STAY POSITIVE UNDER A SPECTRAL GAP ASSUMPTION

Consider a Markov chain (X n) n0 with values in the state space X. Let f be a real function on X and set S 0 = 0, S n = f (X 1) + · · · + f (X n), n 1. Let P x be the probability measure generated by the Markov chain starting at X 0 = x. For a starting point y ∈ R denote by τ y the first moment when the Markov walk (y + S n) n1 becomes non-positive. Under the condition that S n has zero drift, we find the asymptotics of the probability P x (τ y > n) and of the conditional law P x (y + S n · √ n | τ y > n) as n → +∞