Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption

TL;DR

This paper establishes limit theorems for Markov walks conditioned to stay positive, providing asymptotic probabilities and distributions under a spectral gap assumption, extending classical results to Markov-dependent settings.

Contribution

It introduces new asymptotic results for Markov walks conditioned to remain positive, under spectral gap conditions, generalizing classical results for independent processes.

Findings

Asymptotics of $\, ext{P}_x( au_y > n)$ as $n o obreak + obreak \infty$

Conditional law of the scaled walk given it stays positive

Results hold under spectral gap assumption for the Markov chain

Abstract

Consider a Markov chain $(X_n)_{n\geqslant 0}$ with values in the state space $\mathbb X$. Let $f$ be a real function on $\mathbb X$ and set $S_0=0,$ $S_n = f(X_1)+\cdots + f(X_n),$ $n\geqslant 1$. Let $\mathbb P_x$ be the probability measure generated by the Markov chain starting at $X_0=x$. For a starting point $y \in \mathbb R$ denote by $\tau_y$ the first moment when the Markov walk $(y+S_n)_{n\geqslant 1}$ becomes non-positive. Under the condition that $S_n$ has zero drift, we find the asymptotics of the probability $\mathbb P_x ( \tau_y >n )$ and of the conditional law $\mathbb P_x ( y+S_n\leqslant \cdot\sqrt{n} | \tau_y >n )$ as $n\to +\infty.$