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

Abstract EN

Let { ξ i , 1 ≤ i ≤ n } be a sequence of iid U[0, 1]-distributed random variables, and define the uniform empirical process F n ( t ) = n - 1 / 2 ∑ i = 1 n ( I { ξ i ≤ t } - t ) , 0 ≤ t ≤ 1 , F n = s u p 0 ≤ t ≤ 1 | F n ( t ) | .

When the nonnegative function g ( x ) satisfies some regular monotone conditions, it proves that lim ϵ ↘ 0 1 / - l o g ϵ ∑ n = 1 ∞ g ′ ( n ) / g ( n ) E { F n 2 I { ∥ F n ∥ ≥ ϵ g ( n ) } } = π 2 / 6 .