A local limit theorem for random walk defined on a finite Markov chain with absorbing barriers (Q1872588)

Let \((X_n)_{n\geq 0}\) be an ergodic Markov chain on the state space \(\{1,2,\dots,s\}\), and let for all \(j,k\in \{1,2,\dots,s\}\), \((Y^{j,k}_n)_{n\geq 1}\) be independent sequences of i.i.d. \(\{\pm 1\}\)-valued random variables independent of the given Markov chain. The authors derive a local limit theorem for the random walk \((S_n)_{n\geq 0}\) defined by \(S_0\) and \(S_n =S_{n-1} + Y^{X_{n-1},X_n}_n\). The proof is quite straighforward, and the reviewer has the impression that the results hold in a more general setting.