Conditioned local limit theorems for random walks defined on finite Markov chains

Overview & Synopsis

Let (X n) n 0 be a Markov chain with values in a finite state space X starting at X 0 = x ∈ X and let f be a real function defined on X. Set S n = n k=1 f (X k), n 1. For any y ∈ R denote by τ y the first time when y + S n becomes non-positive. We study the asymptotic behaviour of the probability P x (y + S n ∈ [z, z + a] , τ y > n) as n → +∞. We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order n 3/2 and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability P x (τ y = n) as n → +∞.