Limit theorems for Markov walks conditioned to stay positive in the $α$-stable regime under a spectral gap assumption

Let (Xₙ)n≥ 1 be a Markov chain on a measurable state space X, and let Sₙ = ∑ₖ₌₁ⁿ f(Xₖ) be the associated Markov walk. For $y>0$, denote by τy the first time at which y+Sₙ becomes non-positive. Assuming that the centred martingale approximation of Sₙ lies in the domain of attraction of a strictly $α$-stable law with α∈(1,2), and that the transition operator satisfies a spectral-gap condition, we determine the asymptotic behaviour of Pₓ(τy>n). In particular, we show the existence of a strictly positive Q⁺-harmonic function V_α(x,y) such that n1-ρ L(n)\, Pₓ(τy>n) V_α(x,y), where L is slowly varying and $ρ$ is the positivity parameter of the limiting $α$-stable process. We further establish the asymptotic growth of V_α(x,y) as y→∞ and prove a conditional limit theorem: conditionally on \τy>n\, Sₙn1/α L(n) converges in distribution to the $α$-stable meander. These results extend the Gaussian spectral-gap theory of Markov walks to the full stable regime and give the first appearance of stable meanders for Markov additive processes under such assumptions.