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

Abstract

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 → +∞.

The survival probability of critical and subcritical branching processes in finite state space Markovian environment

“…The asymptotic of √ nP i (τ y > n) is given in [13] and using the local limit theorem from [12] we show that the expectation E i (q n | τ y > n) converges to a positive constant. The subcritical case is much more delicate.…”

“…Markov walks conditioned to stay positive. In this section we recall the main results from [13] and [12] for Markov walks conditioned to stay positive. We complement these results by some new assertions which will be used in the proofs.…”

“…The following three assertions deal with the existence of the harmonic function, the limit behaviour of the probability of the exit time and of the law of the random walk y + S n , conditioned to stay positive and are taken from [13].…”

“…and under the assumptions of the paper the random variables η k+1,n are bounded. Our proof is essentially based on three tools: conditioned limit theorems for Markov chains which have been obtained recently in [13] and [12], the exponential change of measure which is defined with the help of the transfer operator, see Guivarc'h and Hardy [15], and the duality for Markov chains which we develop in Section 3.2. Let us first consider the critical case.…”

“…; τ y > l , τ * z > m . (7.8) For any j ∈ X let z 0 (j) ∈ R be the unique real such that (j, z) ∈ supp Ṽ * λ for any z > z 0 and (j, z) / ∈ supp Ṽ * λ for any z < z 0 (see [13] for details on the domain of positivity of the harmonic function). Set z 0 (j) + = max {z 0 (j), 0}.…”