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Second moment convergence rates for uniform empirical processes. (English) Zbl 1202.60046

Summary: Let \(\{U_1,U_2,\dots,U_n\}\) be a sequence of independent and identically distributed \(U[0,1]\)-distributed random variables. Define the uniform empirical process as
\[ \alpha_n(t)=n^{-1/2}\sum^n_{i=1} (I\{U_i\leq t\}-t),\quad 0\leq t\leq 1, \] \(\|\alpha_n\|=\sup_{0\leq t\leq 1}|\alpha_n(t)|\). In this paper, we get the exact convergence rates of weighted infinite series of \(E\|\alpha_n \|^2 I\{\|\alpha_n\|\geq\varepsilon (\log n)^{1/\beta}\}\).

MSC:

60F15 Strong limit theorems
60F99 Limit theorems in probability theory
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