In  Doeblin obtained a central limit for discrete parameter Markov chains with discrete state space. In obtaining this theorem the principal tool is the Doeblin dissection of the sequence of partial sums of a functional of a Markov chain into a random sum of independent, identically distributed random variables. Indeed, given this dissection, the remainder of Doeblin's proof is, in essence, a proof of a random central limit theorem. Although Doeblin does not state a stable limit theorem for Markov chains, at the end of the paper he observes that the dissection should also be of use in obtaining such theorems, and comments on a possible method of obtaining theorems of this nature. In this paper three stable limit theorems for Markov chains together with the appropriate solidarity theorems are obtained depending on whether the index $\alpha$ of the limiting distribution is $< 1, = 1, or > 1$. The principal tools used in obtaining these theorems are the Doeblin dissection, a well known result concerning the rate of growth of the sequence of norming coefficients of a random variable in the domain of attraction of a stable law, and a random stable limit theorem , .
"A Stable Limit Theorem for Markov Chains." Ann. Math. Statist. 40 (4) 1467 - 1473, August, 1969. https://doi.org/10.1214/aoms/1177697517