On the central and local limit theorem for martingale difference sequences

Details On the central and local limit theorem for martingale difference sequences On the central and local limit theorem for martingale difference sequences

2004-02-28

arxiv.org/abs/math/0403008v1

Mathematics

Probability

Accepte pour publication dans Stochastics and Dynamics

Scientific paper

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

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