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.
Publié le : 2004-02-28
Classification:
measure-preserving transformation.,
measure-preserving transformation,
rate of convergence,
mixing,
martingale difference sequence,
local limit theorem,
central limit theorem,
60F99, 28D05,
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
@article{hal-00001224,
author = {El Machkouri, Mohamed and Volny, Dalibor},
title = {On the central and local limit theorem for martingale difference sequences},
journal = {HAL},
volume = {2004},
number = {0},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00001224}
}
El Machkouri, Mohamed; Volny, Dalibor. On the central and local limit theorem for martingale difference sequences. HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00001224/