Lower bounds for the constants in non-uniform estimates of the rate of convergence in the CLT

Abstract

We conduct a comparative analysis of the constants in the Nagaev-Bikelis and Bikelis-Petrov inequalities which establish non-uniform estimates of the rate of convergence in the central limit theorem for sums of independent random variables possessing finite absolute moments of order $2+\delta$ with $\delta \in [0,1]$. We provide lower bounds for the above constants and also for the constants in the structural improvements of Nagaev-Bikelis' inequality. The lower bounds in Nagaev-Bikelis' inequality and it's structural improvements are given in dependence on $\delta$ and a structural parameter $s$ as well as uniform with respect to both $\delta$ and $s$. Lower bounds for the constants in Nagaev-Bikelis' with $\delta <1$ and Bikelis-Petrov's inequalities are presented for the first time.