Renewal Theory for Functionals of a Markov Chain with General State Space

We prove an analogue of Blackwell's renewal theorem or the "key renewal theorem" and the existence of the limit distribution of the residual waiting time in the following setup: $X_0, X_1, \cdots$ is a Markov chain with separable metric state space and $u_0, u_1, \cdots$ is a sequence of random variables, such that the conditional distribution of $u_i$, given all $X_j$ and $u_l, l \neq i$, depends on $X_i$ and $X_{i+1}$ only. Here the $V_n \equiv \sum^{n-1}_0 u_i, n \geqq 1$, take the role of the partial sums of independent identically distributed random variables in ordinary renewal theory. E.g. the key renewal theorem in this setup states that $\lim_{t\rightarrow\infty} E\{\sum^\infty_{n=0} g(X_n, t - V_n)\mid X_0 = x\}$ exists for suitable $g(\bullet, \bullet)$, and is independent of $x$.