Improved Berry-Esseen Bound for Rademacher Sum

Let X=\sum\limits_i=1^n a_i \xi_i be a Rademacher sum with Var (X) = 1 and Z be a standard normal random variable. This paper concerns the upper bound of |\mathsfP(X \leqslant x)-\mathsfP(Z \leqslant x)| for any x ∈ \mathbbR . Using the symmetric properties and R software, this paper gets the following improved BerryEsseen type bound under some conditions, |\mathsfP(X \leqslant x)-\mathsfP(Z \leqslant x)|\leqslant \mathsfP\left(Z \in\left(0, a_1\right)\right), \forall x \in \mathbbR, which is one of the modified conjecture proposed by Nathan K. and Ohad K.