A refined randomized concentration inequality

Abstract

Let ξ1,ξ2,…,ξn be independent random variables with Eξi=0 and ∑i=1nEξi2=1 and let Δ=Δ(ξ1,…,ξn). Set W=∑i=1nξi. In this note we prove that |P(W≤Δ)−EΦ(Δ)|≤36∑i=1nEξi2min(1,|ξi|)+24∑i=1n‖ξi‖2‖Δ−Δ(i)‖2, where Δ(i) is any random variable that doesn’t depend on ξi. The result leads to a refined Berry–Esseen inequality for non-linear statistics W+Δ as well as a refined concentration inequality for P(W≤Δ2)−P(W≤Δ1).