On certain external property of the symmetric Bernoulli distribution (Q1909851)

The paper is devoted to the proof of the inequality \[ \int^{x+2}_x P\Biggl(\Biggl|\sum^n_{i=1}\xi_i\Biggr|\geq t\Biggr)dt\leq \int^{x+2}_x P\Biggl(\Biggl|\sum^n_{i=1}\varepsilon_i\Biggr|\geq t\Biggr)dt \] in which \(\xi_i\) are independent symmetric \(|\xi_i|\leq 1\), generally not identically distributed, while \(\varepsilon_i\) are independent all taking the values \(1\) and \(-1\) with probabilities \(1/2\). The proof starts by decomposing \(\xi_i\) into integrals of symmetric two valued ones. There are some comments, remarks about related inequalities (some of them, with factor 2 in the right member), a corollary and a conjecture, stating that the inequality is true if we replace the hypothesis of symmetry by ``null mean''.