Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem

Abstract

Following (Shevtsova, 2013) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen’s, Rozovskii’s, and Wang–Ahmad’s inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones.

Keywords

• central limit theorem
• Lindeberg’s theorem
• normal approximation
• asymptotically exact constant
• asymptotically best constant
• uniform distance