Convergence rates are derived in central limit theorems for martingale difference arrays. The rates depend heavily on the behavior of the conditional variances and on moment conditions. It is also shown that the rates which are obtained are the exact ones under the stated conditions.
"Exact Convergence Rates in Some Martingale Central Limit Theorems." Ann. Probab. 10 (3) 672 - 688, August, 1982. https://doi.org/10.1214/aop/1176993776