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Large Deviations Application to Billingsley's Example

27 May 2009
R. Liptser
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Abstract

We consider a classical model related to an empirical distribution function Fn(t)=1n∑k=1nI{ξk≤t} F_n(t)=\frac{1}{n}\sum_{k=1}^nI_{\{\xi_k\le t\}}Fn​(t)=n1​∑k=1n​I{ξk​≤t}​ of (ξk)i≥1(\xi_k)_{i\ge 1}(ξk​)i≥1​ -- i.i.d. sequence of random variables, supported on the interval [0,1][0,1][0,1], with continuous distribution function F(t)=P(ξ1≤t)F(t)=\mathsf{P}(\xi_1\le t)F(t)=P(ξ1​≤t). Applying ``Stopping Time Techniques'', we give a proof of Kolmogorov's exponential bound \mathsf{P}\big(\sup_{t\in[0,1]}|F_n(t)-F(t)|\ge \varepsilon\big)\le \text{const.}e^{-n\delta_\varepsilon} conjectured by Kolmogorov in 1943. Using this bound we establish a best possible logarithmic asymptotic of \mathsf{P}\big(\sup_{t\in[0,1]}n^\alpha|F_n(t)-F(t)|\ge \varepsilon\big) with rate 1n1−2α \frac{1}{n^{1-2\alpha}} n1−2α1​ slower than 1n\frac{1}{n}n1​ for any α∈(0,1/2)\alpha\in\big(0,{1/2}\big)α∈(0,1/2).

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