Limit theorems for sums of chain-dependent processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.