Second Moment Convergence Rates for Uniform Empirical Processes

Let {U-1, U-2,..., U-n} be a sequence of independent and identically distributed U [0, 1]-distributed random variables. Define the uniform empirical process as an alpha(n)(t) = n(-1/2) Sigma(n)(i=1) (I{Ui <= t} -t), 0 <= t <= 1, parallel to alpha(n)parallel to = sup(0 <= t <= 1)|alpha(n)(t)|. In this paper, we get the exact convergence rates of weighted infinite series of E parallel to alpha(n)parallel to I-2{parallel to alpha(n)parallel to >=epsilon(log n)(1/beta)}.