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

Abstract

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 ${\tau }_y$ 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 $\alpha$-stable law with $\alpha \in (1,2)$, and that the transition operator satisfies a spectral-gap condition, we determine the asymptotic behaviour of $P_x({\tau }_y>n)$. In particular, we show the existence of a strictly positive $Q^{+}$-harmonic function $V_{\alpha }(x,y)$ such that $$n^{1-\rho }L(n)P_x({\tau }_y>n)⟶V_{\alpha }(x,y),$$ where $L$ is slowly varying and $\rho$ is the positivity parameter of the limiting $\alpha$-stable process. We further establish the asymptotic growth of $V_{\alpha }(x,y)$ as $y\to \infty$ and prove a conditional limit theorem: conditionally on $\{{\tau }_y>n\}$, $$\frac{S_n}{n^{1/\alpha }L(n)}$$ converges in distribution to the $\alpha$-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.