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LDP application for Billingsley's example

27 May 2009
R. Liptser
ArXiv (abs)PDFHTML
Abstract

We consider a classical model discussed in Theorem 16.4 (Billingsley \cite{Bil}) concerning to an empirical distribution function F_n(t)=\frac{1}{n}\sum_{k=1}^nI_{\{\xi_k\le 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 F(t)=P(ξ1≤t)F(t)=\mathsf{P}(\xi_1\le t)F(t)=P(ξ1​≤t) the continuous distribution function. We give a proof of Kolmogorov's exponential estimate \mathsf{P}\Big(\sup_{t\in[0,1]}|F_n(t)-F(t)|\ge \varepsilon\Big) \le 2\exp\Big\{-n[\varepsilon/8\log(1+\varepsilon^2/32)-\varepsilon/8 +(4/\varepsilon)\log(1+\varepsilon^2/32\Big)]\} with the help of which jointly with the large deviations technique, we establish a logarithmic asymptotic: for any T∈[F−1(1/2),1)T\in[F^{-1}({1/2}),1)T∈[F−1(1/2),1) and any α∈(0,1/2)\alpha\in\big(0,{1/2}\big)α∈(0,1/2): \lim_{n\to\infty}\frac{1}{n^{1-2\alpha}} \log\mathsf{P}\bigg(\sup_{t\in[0,T]}n^\alpha\Big|F_n(t)-F(t)\Big|\ge \varepsilon\bigg)=-2\varepsilon^2.

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