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

Abstract

Let $$(X_n)_{n\geqslant 0}$$ 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 $$n\rightarrow +\infty .$$ 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 $$n\rightarrow +\infty $$.