Lower Bounds for the Constants in Non-Uniform Estimates of the Rate of Convergence in the CLT

We conduct a comparative analysis of the constants in the Nagaev–Bikelis and Bikelis–Petrov inequalities that 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 + δ with δ ∈ [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 its structural improvements are given as a function of δ and a structural parameter s as well as uniform with respect to both δ and s. Lower bounds for the constants in Nagaev–Bikelis’ with δ < 1 and Bikelis–Petrov’s inequalities are presented for the first time. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.