Estimates for the accuracy of coupling in the central limit theorem (Q1358027)

For independent centered random variables \(\xi_1,\ldots,\xi_N\) the author studies the difference \(\Delta={}S/B-Z{}\) where \(S=\sum_{j=1}^N X_j, 0<B^2=\sum_{j=1}^N\text{var} \xi_j<\infty\) and \(Z\) has a standard normal law. More exactly \(S\) and \(Z\) are constructed on some probability space. Under certain conditions the upper bound for \({\mathbf E}\Delta^{\alpha}\exp{(\lambda\Delta/8)}\) is obtained in terms of the modification of Lyapunov's ratio (here \(\alpha\geq 1\) and \(\lambda>0\) is chosen in an appropriate manner). Several estimates for other functions of \(\Delta\) are also established.