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- W2806584646 abstract "We consider a classical model related to an empirical distribution function $ F_n(t)=frac{1}{n}sum_{k=1}^nI_{{xi_kle t}}$ of $(xi_k)_{ige 1}$ -- i.i.d. sequence of random variables, supported on the interval $[0,1]$, with continuous distribution function $F(t)=mathsf{P}(xi_1le t)$. Applying ``Stopping Time Techniques'', we give a proof of Kolmogorov's exponential bound $$ mathsf{P}big(sup_{tin[0,1]}|F_n(t)-F(t)|ge varepsilonbig)le text{const.}e^{-ndelta_varepsilon} $$ conjectured by Kolmogorov in 1943. Using this bound we establish a best possible logarithmic asymptotic of $$ mathsf{P}big(sup_{tin[0,1]}n^alpha|F_n(t)-F(t)|ge varepsilonbig) $$ with rate $ frac{1}{n^{1-2alpha}} $ slower than $frac{1}{n}$ for any $alphainbig(0,{1/2}big)$." @default.
- W2806584646 created "2018-06-13" @default.
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- W2806584646 date "2009-05-27" @default.
- W2806584646 modified "2023-09-26" @default.
- W2806584646 title "LDP application for Billingsley's example" @default.
- W2806584646 hasPublicationYear "2009" @default.
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