Esseen–Rozovskii Type Estimates for the Rate of Convergence in the Lindeberg Theorem

Abstract

We present structural improvements of Esseen’s (1969) and Rozovskii’s (1974) estimates for the rate of convergence in the Lindeberg theorem and also compute the appearing absolute constants. We introduce the asymptotically exact constants in the constructed inequalities and obtain upper bounds for them. We analyze the values of Esseen’s, Rozovskii’s, and Lyapunov’s fractions, compare them pairwise, and provide some extremal distributions. As an auxiliary statement, we prove a sharp inequality for the quadratic tails of an arbitrary distribution (with finite second-order moment) and its convolutional symmetrization.