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 \(E\|\alpha_n \|^2 I\{\|\alpha_n\|\geq\varepsilon (\log n)^{1/\beta}\}\).


60F15 Strong limit theorems
60F99 Limit theorems in probability theory
Full Text: DOI EuDML


[1] Hsu, PL; Robbins, H, Complete convergence and the law of large numbers, Proceedings of the National Academy of Sciences of the United States of America, 33, 25-31, (1947) · Zbl 0030.20101
[2] Erdös, P, On a theorem of hsu and robbins, Annals of Mathematical Statistics, 20, 286-291, (1949) · Zbl 0033.29001
[3] Chen, R, A remark on the tail probability of a distribution, Journal of Multivariate Analysis, 8, 328-333, (1978) · Zbl 0376.60033
[4] Gut, A; Spătaru, A, Precise asymptotics in the baum-katz and Davis laws of large numbers, Journal of Mathematical Analysis and Applications, 248, 233-246, (2000) · Zbl 0961.60039
[5] Heyde, CC, A supplement to the strong law of large numbers, Journal of Applied Probability, 12, 173-175, (1975) · Zbl 0305.60008
[6] Chow, YS, On the rate of moment convergence of sample sums and extremes, Bulletin of the Institute of Mathematics, 16, 177-201, (1988) · Zbl 0655.60028
[7] Liu, W; Lin, Z, Precise asymptotics for a new kind of complete moment convergence, Statistics & Probability Letters, 76, 1787-1799, (2006) · Zbl 1104.60015
[8] Fu, K-A, Asymptotics for the moment convergence of [inlineequation not available: see fulltext.]-statistics in LIL, Journal of Inequalities and Applications, 2010, 8, (2010) · Zbl 1200.62011
[9] Yan, JG; Su, C, Precise asymptotics of [inlineequation not available: see fulltext.]-statistics, Acta Mathematica Sinica, 50, 517-526, (2007) · Zbl 1141.60329
[10] Pang, T-X; Zhang, L-X; Wang, JF, Precise asymptotics in the self-normalized law of the iterated logarithm, Journal of Mathematical Analysis and Applications, 340, 1249-1262, (2008) · Zbl 1140.60023
[11] Zang, Q-P, A limit theorem for the moment of self-normalized sums, Journal of Inequalities and Applications, 2009, 10, (2009) · Zbl 1186.60029
[12] Zhang, Y; Yang, X-Y, Precise asymptotics in the law of the iterated logarithm and the complete convergence for uniform empirical process, Statistics & Probability Letters, 78, 1051-1055, (2008) · Zbl 1140.60316
[13] Csörgő M, Révész P: Strong Approximations in Probability and Statistics, Probability and Mathematical Statistics. Academic Press, New York, NY, USA; 1981:284. · Zbl 0539.60029
[14] Stout WF: Almost Sure Convergence. Academic Press, New York, NY, USA; 1974:x+381. Probability and Mathematical Statistics, Vol. 2 · Zbl 0321.60022
[15] Kiefer, J; Wolfowitz, J, On the deviations of the empiric distribution function of vector chance variables, Transactions of the American Mathematical Society, 87, 173-186, (1958) · Zbl 0088.11305
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