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

Let $(X_n)_{n\geq 0}$ 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 $S_n=\sum_{k=1}^{n} f(X_k)$, $n\geqslant 1$. 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 $n^{3/2}$ and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability $\mathbb P_x \left( \tau_y = n \right)$ as $n\to+\infty$.