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

TL;DR

This work analyzes Markov walks conditioned to stay positive when increments lie in the α-stable domain of attraction under a spectral-gap framework. It develops a martingale-based decomposition and a stable strong approximation to a strictly α-stable Lévy process, yielding sharp asymptotics for the exit probability P_x(τ_y>n) via a strictly positive $Q^+$-harmonic function $V_α(x,y)$. The main results establish the tail behavior P_x(τ_y>n) ~ V_α(x,y)/(n^{1-ρ} L(n)) and show that, conditioned on survival, the normalized walk converges to the α-stable meander; additionally, V_α(x,y) grows like h(x) y^{α(1-ρ)} as y→∞. These findings extend Gaussian spectral-gap theory to the stable regime and introduce stable meanders for Markov additive processes, broadening the understanding of conditioned Markov walks in heavy-tailed settings.

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 $τ_y$ 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.

Theorems & Definitions (15)

• Definition 2.1
• Lemma 3.1: Control of the Remainder
• Remark
• Lemma 3.2: Martingale Difference Moments
• Remark
• Proposition 4.1: Stable Strong Approximation
• proof
• Lemma 4.2: Comparison of Exit Times
• proof
• Proposition 4.3: Stable Exit Tail