CENTRAL AND LOCAL LIMIT THEOREMS IN MARKOV DEPENDENT RANDOM VARIABLES

We consider an irreducible and aperiodic Markov chain {kn}n=o over the finite state space E = {1, ... , p} with positive regular transition matrix P = {pj} and additive component {Un} such that {Sn} = {(kn, Un)} is also a Markov chain over the state space E = E x R. We prove a central and a local limit theorem for this chain when the probability density functions of {Sn}, conditional on the first and the last states of {k n}n = 0, exist.