## 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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### References:

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