We prove the following conjecture, due to Tomaszewski (1986): Let X=∑i=1naixi, where ∑iai2=1 and each xi is a uniformly random sign. Then Pr⁡[|X|≤1]≥1/2. Our main novel tools are local concentration inequalities an improved Berry-Esseen inequality for Rademacher sums.

* Correspondence: lym1019@163. com Department of Mathematics, Shangrao Normal University, Shangrao 334001, China Full list of author information is available at the end of the article Abstract Consider the semiparametric regression model Yi = xi b + g (ti) + εi, i = 1, . . . , n, where the linear process errors εi = ∑∞ j=−∞ ajei−j with ∑∞ j=−∞ ∣∣aj∣∣ < ∞ , and {ei} are identically distributed a...