We consider a classical model related to an empirical distribution function of -- i.i.d. sequence of random variables, supported on the interval , with continuous distribution function . 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 slower than for any .
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