The rate of convergence in the central limit theorem via Zolotarev probability metric

Abstract

The central limit theorem is a well-known theorem in probability theory. It is the theoretical basis for constructing statistical problems such as parameter estimation problems and statistical hypothesis testing, etc. The main aim of this article is to estimate the upper bound in the central limit theorem for independent but not necessarily identically distributed random variables under Lyapunov's conditions via the Zolotarev probability metric. The obtained result is the rate of convergence in the central limit theorem for independent random variables. In the case of independent identically distributed random variables will be concluded as a direct corollary. The Zolotarev probability metric is the main research tool in this paper since it is an ideal metric of order s > 0. Furthermore, the Zolotarev probability metric may be compared with well-known metrics like the Kolmogorov metric, total variation metric, the Levy-Prokhorov metric, and the metric based on the Trotter operator, etc.