Large Deviations Application to Billingsley's Example

Abstract

We consider a classical model related to an empirical distribution function of (ξ _k ) _i≥1 – i.i.d. sequence of random variables, supported on the interval [0, 1], with continuous distribution function . Applying “Stopping Time Techniques”, we give a proof of Kolmogorov's exponential bound conjectured by Kolmogorov in 1943. Using this bound we establish a best possible logarithmic asymptotic of with rate slower than for any .

Keywords:

• Empirical distribution
• Large deviations principle
• Stopping time

Subject Classification:

• 60F10
• 60J27

Notes

Recommended by A. G. Tartakovsky