Summary
Let \(U_n (t) = n^{\tfrac{1}{2}} (\Gamma _n (t) - t), 0 \leqq t \leqq 1\), denote the uniform empirical process based on the first n of a sequence ξ 1, ξ 2, ... of iid uniform (0,1) random variables where \(\Gamma _n (t) = n^{ - 1} \sum\limits_{i = 1}^n {1_{[0,t]} } (\xi _i )\) is the empirical distribution function. The oscillation modulus of U n is defined by
, and the Lipschitz-1/2 modulus of U n is defined by
Strong limit theorems are presented for both Ω n(a) and \(\tilde \omega _n (a)\) with a= a n→0 at various rates. For ‘short’ intervals with a n=cn −1logn, c>0, the results are related to Erdos-Rényi strong laws of large numbers; at the other extreme, for ‘long’ intervals with a n=1/(logn)c, c>0, the results are related to laws of the iterated logarithm for U n.
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Research supported in part by the National Science Foundation under MCS 81-02568
Research supported in part by the National Science Foundation under MCS 81-02731
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Mason, D.M., Shorack, G.R. & Wellner, J.A. Strong limit theorems for oscillation moduli of the uniform empirical process. Z. Wahrscheinlichkeitstheorie verw Gebiete 65, 83–97 (1983). https://doi.org/10.1007/BF00534996
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DOI: https://doi.org/10.1007/BF00534996