On the Convergence Rate in a Local Renewal Theorem for a Random Markov Walk

Suppose that a sequence \(\{X_n\}_{n\ge 0}\) of random variables is a homogeneous irreducible Markov chain with finite set of states. Let \(\xi_n\) , \(n\in\mathbb{N}\) , be random variables defined on the chain transitions.

The renewal function \(u_k:=\sum_{n=0}^{+\infty} \mathsf P(S_n=k), \qquad k\in\mathbb{N},\) where \(S_0:=0\) and \(S_n:=\xi_1+\dots + \xi_n\) , \(n\in\mathbb{N}\) , is introduced. It is shown that this function converges to its limit at an exponential rate, and an explicit description of the exponent is given.