Central limit theorem and Diophantine approximations

Abstract

Let $F_n$ denote the distribution function of the normalized sum $Z_n=(X_1+\cdots +X_n)/\sigma \sqrt{n}$ of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of $F_n$ to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of $F_n$ by the Edgeworth corrections (modulo logarithmically growing factors in $n$) are given in terms of the characteristic function of $X_1$. Particular cases of the problem are discussed in connection with Diophantine approximations.