Precise Asymptotics on Second-Order Complete Moment Convergence of Uniform Empirical Process

Let{ξi,1≤i≤n}be a sequence of iidU[0, 1]-distributed random variables, and define the uniform empirical processFn(t)=n-1/2∑i=1n‍(I{ξi≤t}-t),0≤t≤1,Fn=sup0≤t≤1|Fn(t)|. When the nonnegative functiong(x)satisfies some regular monotone conditions, it proves thatlimϵ↘0⁡1/-logϵ∑n=1∞g′(n)/g(n)E{Fn2I{∥Fn∥≥ϵg(n)}}=π2/6.