A local limit theorem for random walk defined on a finite Markov chain with absorbing barriers

Abstract

Let {ξ_n}_n≥0 denote an ergodic Markov chain with a finite state space Ξ={1, 2, ..., s}. For each j, k∈Ξ, let {Y_n^jk}_n≥1 be a sequence of i.i.d. {−1, 1}-valued random variables which are independent of {ξ_n}. We define the process {S_n}_{n≥ 0} by S₀=0 and S_n=S_n−1+Y_n^ξ_n−1ξ_n for n≥ 1. Let a be a positive integer. We denote by T_x the first exit time of the process from the interval [−x, a−x] for each x=0, 1, ..., a. We give an asymptotic behaviour of the transition functions P_jk⁽ⁿ⁾(x, y)=P{x+S_n=y; T_x>n; ξ_n=k|ξ₀=j} as n→∞ for each x, y∈[0, a] and all j, k∈Ξ.