On the central and local limit theorem for martingale difference sequences

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Let (Ω,, μ) be a Lebesgue space and T an ergodic measure preserving automorphism on Ω with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on Ω 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.