Conditioned local limit theorems for random walks defined on finite Markov chains

Let $(X_n{)}_{n\ge 0}$ be a Markov chain with values in a finite state space $\mathbb{X}$ starting at $X_0=x\in \mathbb{X}$ and let $f$ be a real function defined on $\mathbb{X}$. Set $S_n={\sum }_{k=1}^{n}f(X_k)$, $n⩾1$. For any $y\in \mathbb{R}$ denote by ${\tau }_y$ the first time when $y+S_n$ becomes non-positive. We study the asymptotic behaviour of the probability $\mathbb P_x \left( y+S_{n} \in [z,z+a] ,,, \tau_y > n \right)$as$n\to+\infty.$ We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order $n^{3\mathrm{/}2}$ and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability ${\mathbb{P}}_x({\tau }_y=n)$ as $n\to +\mathrm{\infty }$.