On asymptotic expansions for distribution functions of sums of independent random variables

Let $\{X_j\}$ be a sequence of independent identically distributed random variables with zero means and unit variances and let $F_n(x)$ be the distribution function of the sum $\frac1{\sqrt n}\sum_{j=1}^nX_j$. Asymptotic expansions of the function $F_n(x)$ are given which are more general than the classic expansion (0.1). We study also the asymptotic behaviour of the remainder in (0.1).