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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.16124 |
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Table of Contents:
- Let $(X_n)_{n\ge 1}$ be a Markov chain on a measurable state space $X$, and let $S_n = \sum_{k=1}^n f(X_k)$ be the associated Markov walk. For $y>0$, denote by $τ_y$ the first time at which $y+S_n$ becomes non-positive. Assuming that the centred martingale approximation of $S_n$ lies in the domain of attraction of a strictly $α$-stable law with $α\in(1,2)$, and that the transition operator satisfies a spectral-gap condition, we determine the asymptotic behaviour of $P_x(τ_y>n)$. In particular, we show the existence of a strictly positive $Q^+$-harmonic function $V_α(x,y)$ such that $$n^{1-ρ} L(n)\, P_x(τ_y>n) \longrightarrow V_α(x,y),$$ 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 $y\to\infty$ and prove a conditional limit theorem: conditionally on $\{τ_y>n\}$, $$\frac{S_n}{n^{1/α} L(n)}$$ 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.