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 indecomposable Markov chain with finite set of states. Let $\xi_n$, $n\in\mathbb{N}$, be random variables defined on the chain transitions.
The reconstruction 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 with exponential rate, and an explicit description of the exponent is given.