We prove the following central limit theorems for sums of mutually dependent random vector variables: Given that a sequence of random vector variables satisfies a certain type of decoupling condition (and two milder restrictions), we present a Lindeberg-Feller condition which we show to be both necessary and sufficient for central limit behavior. The decoupling condition and one of the two milder conditions is then applied to a Markov process with stationary transition mechanism.
W. J. Cocke. "Central Limit Theorems for Sums of Dependent Vector Variables." Ann. Math. Statist. 43 (3) 968 - 976, June, 1972. https://doi.org/10.1214/aoms/1177692559