Toward the best constant factor for the Rademacher-Gaussian tail comparison

Abstract

Let S_n:=a_1\vp_1+...+a_n\vp_n, where \vp_1,...,\vp_n are independent Rademacher random variables (r.v.&#39;s) and a_1,...,a_n are any real numbers such that a_1^2+...+a_n^2=1. Let Z be a standard normal r.v. It is proved that the best constant factor c in inequality ¶(S_n>x) \leq c¶(Z>x) for all x in \R is between two explicitly defined absolute constants c_1 and c_2 such that c_1<c_2 \approx 1.01c_1.