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In this work we derive a variant of the classic Glivenko-Cantelli Theorem,which asserts uniform convergence of the empirical Cumulative DistributionFunction (CDF) to the CDF of the underlying distribution. Our variant allowsfor tighter convergence bounds for extreme values of the CDF.  We apply our bound in the context of revenue learning, which is awell-studied problem in economics and algorithmic game theory. We derivesample-complexity bounds on the uniform convergence rate of the empiricalrevenues to the true revenues, assuming a bound on the $k$th moment of thevaluations, for any (possibly fractional) $k>1$.  For uniform convergence in the limit, we give a complete characterization anda zero-one law: if the first moment of the valuations is finite, then uniformconvergence almost surely occurs; conversely, if the first moment is infinite,then uniform convergence almost never occurs.

Submultiplicative Glivenko-Cantelli and Uniform Convergence of Revenues