On the central and local limit theorem for martingale difference sequences

Let $(\mathrm{\Omega },\text{A},\mu )$ be a Lebesgue space and $T$ an ergodic measure preserving automorphism on $\mathrm{\Omega }$ with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on $\mathrm{\Omega }$ with a common non-degenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A similar result is established for martingale difference sequences with densities provided the entropy is infinite. In addition, the martingale difference sequence may be chosen to be strongly mixing.