On the Error-Bound in the Nonuniform Version of Esseen's Inequality in the L p -Metric

The aim of this paper is to investigate the known nonuniform version of Esseen's Inequality in the L P -metric, p ≥ 1, to get a numerical bound for the appearing constant L, see (2.1). For a long time the results given by several authors constate the impossibility of (2.1) in the most interesting case δ = 1, because the effect was observed, where 2 + δ, 0 < δ ≤ 1, is the order of the assumed moments of the considered independent random variables X k , k = 1, 2, ..., n. Again making use of the method of conjugated distributions, cp., 5,11,19,26 we improve the well-known technique to show in the most interesting case δ = 1 the finiteness of the absolute constant L and to prove where . In the case δ ∈(0,1) we only give the analytical structure of L but omit numerical calculations. Finally an example on normal approximation of sums of l 2 -valued random elements demonstrates the application of the nonuniform mean central limit bounds obtained here.

VIVO