Let (Xₙ) ₍ ₁ be a Markov chain on a measurable state space X, and let Sₙ = ₊=₁ⁿ f (Xₖ) be the associated Markov walk. For y>0, denote by τᵧ 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 (τᵧ>n). In particular, we show the existence of a strictly positive Q^+-harmonic function V_α (x, y) such that n^1-ρ L (n) \, Pₓ (τᵧ>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 \τᵧ>n\, Sₙn^{1/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.
Zhao et al. (Thu,) studied this question.