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