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

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 = ∑ k = 1 n 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 → + ∞ .