  Sums of independent, bounded random variables concentrate around their
expectation approximately as well a Gaussian of the same variance. Well known
results of this form include the Bernstein, Hoeffding, and Chernoff
inequalities and many others. We present an alternative proof of these tail
bounds based on what we call a stability argument, which avoids bounding the
moment generating function or higher-order moments of the distribution. Our
stability argument is inspired by recent work on the generalization properties
of differential privacy and their connection to adaptive data analysis (Bassily
et al., STOC 2016).
