Large Deviations Application to Billingsley's Example

TL;DR

This paper applies large deviations techniques to empirical distribution functions, providing a proof of Kolmogorov's exponential bound and establishing optimal asymptotic rates for deviations.

Contribution

It introduces a novel application of stopping time techniques to prove Kolmogorov's bound and derives the best possible logarithmic asymptotics for empirical distribution deviations.

Findings

Proof of Kolmogorov's exponential bound using stopping time techniques

Establishment of optimal logarithmic asymptotics for empirical distribution deviations

Identification of the rate of decay slower than 1/n for certain deviations

Abstract

We consider a classical model related 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 continuous distribution function $F(t)=\mathsf{P}(\xi_1\le t)$. Applying ``Stopping Time Techniques'', we give a proof of Kolmogorov's exponential bound $$ \mathsf{P}\big(\sup_{t\in[0,1]}|F_n(t)-F(t)|\ge \varepsilon\big)\le \text{const.}e^{-n\delta_\varepsilon} $$ 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 $ \frac{1}{n^{1-2\alpha}} $ slower than $\frac{1}{n}$ for any $\alpha\in\big(0,{1/2}\big)$.