It is shown that a method recently developed by Bolthausen permits an extension (up to a logarithmic factor) of an estimate of the rate of convergence in the martingale central limit theorem due to Heyde and Brown.
"A Note on the Rate of Convergence in the Martingale Central Limit Theorem." Ann. Probab. 12 (2) 635 - 639, May, 1984. https://doi.org/10.1214/aop/1176993311