Some bounds on the rate of convergence in the CLT for martingales. I

This paper concerns rates of convergence in the central limit theorem (CLT) for the random variables $S_{n}=\sum_{1}^{n}X_{m}$, where $X_{m}$ are martingale-differences. It is known that in the general case one cannot hope for a rate better than $O(n^{-1/8})$ even if the third moments are finite. If the conditional variances satisfy $\mathsf{E}\{X_{m}^2\mid X_{1}\ldots X_{m-1}\}=\mathsf{E}X_{m}^2$, the rate in general is no better than $O(n^{-1/4}),$ while in the independency case it is of the order $O(n^{-1/2})$. This paper contains a bound which covers simultaneously the cases mentioned as well as some intermediate cases. The bound is presented in terms of some dependency characteristics reflecting the influence of different factors on the rate.