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.