Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption

Consider a Markov chain (Xn)n.0 with values in the state space X. Let f be a real function on X and set Sn=.ni=1f(Xi), n.1. Let Px be the probability measure generated by the Markov chain starting at X0=x. For a starting point y.R, denote by .y the first moment when the Markov walk (y+Sn)n.1 becomes nonpositive. Under the condition that Sn has zero drift, we find the asymptotics of the probability Px(.y>n) and of the conditional law Px(y+Sn..n.....y>n) as n.+..