Consider a Markov chain (X n ) n 0 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 ) n 1 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 → +∞.

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 → +∞.

Consider the real Markov walk S n = X 1 + · · · + X n with increments (X n ) n 1 defined by a stochastic recursion starting at X 0 = x. For a starting point y > 0 denote by τ y the exit time of the process (y + S n ) n 1 from the positive part of the real line. We investigate the asymptotic behaviour of the probability of the event τ y n and of the conditional law of y + S n given τ y n as n → +∞.

No abstract

No abstract