Large Deviations Application to Billingsley's Example

Abstract

We consider a classical model related to an empirical distribution function $F_n(t)=\frac{1}{n}{\sum }_{k=1}^n{I}_{\{{\xi }_k\le t\}}$ of $({\xi }_k{)}_{i\ge 1}$ -- i.i.d. sequence of random variables, supported on the interval $[0,1]$, with continuous distribution function $F(t)=\mathsf{P}({\xi }_1\le t)$. Applying ``Stopping Time Techniques'', we give a proof of Kolmogorov's exponential bound $$\mathsf{P}(\underset{t\in [0,1]}{\sup }|F_n(t)-F(t)|\ge \epsilon )\le \text{const.}{e}^{-n{\delta }_{\epsilon }}$$ conjectured by Kolmogorov in 1943. Using this bound we establish a best possible logarithmic asymptotic of $$\mathsf{P}(\underset{t\in [0,1]}{\sup }{n}^{\alpha }|F_n(t)-F(t)|\ge \epsilon )$$ with rate $\frac{1}{n^{1-2\alpha }}$ slower than $\frac{1}{n}$ for any $\alpha \in (0,1/2)$.

60F10, 60J27