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

Abstract

Consider a Markov chain $(X_n{)}_{n⩾0}$ with values in the state space $\mathbb{X}$. Let $f$ be a real function on $\mathbb{X}$ and set $S_0=0,$ $S_n=f(X_1)+\cdots +f(X_n),$ $n⩾1$. 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 $(y+S_n{)}_{n⩾1}$ 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 ${\mathbb{P}}_x(y+S_n⩽\cdot \sqrt{n}|{\tau }_y>n)$ as $n\to +\infty .$

Figure 1 corrected