  A refinement of Bennett's inequality is introduced which is strictly tighter
than the classical bound. The new bound establishes the convergence of the
average of independent random variables to its expected value. It also
carefully exploits information about the potentially heterogeneous mean,
variance, and ceiling of each random variable. The bound is strictly sharper in
the homogeneous setting and very often significantly sharper in the
heterogeneous setting. The improved convergence rates are obtained by
leveraging Lambert's W function. We apply the new bound in a portfolio
optimization setting to allocate a budget across investments with heterogeneous
returns.
