LDP application for Billingsley's example

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 $(\xi_k)_{i\ge 1}$ - i.i.d. sequence of random variables, supported on the interval $[0,1]$, with $F(t)=\mathsf{P}(\xi_1\le t)$ 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 $T\in[F^{-1}({1/2}),1)$ and any $\alpha\in\big(0,{1/2}\big)$: \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.