  Under the assumption that the distribution of a nonnegative random variable
$X$ admits a bounded coupling with its size biased version, we prove simple and
strong concentration bounds. In particular the upper tail probability is shown
to decay at least as fast as the reciprocal of a Gamma function, guaranteeing a
moment generating function that converges everywhere. The class of infinitely
divisible distributions with finite mean, whose L\'evy measure is supported on
an interval contained in $[0,c]$ for some $c < \infty$, forms a special case in
which this upper bound is logarithmically sharp. In particular the asymptotic
estimate for the Dickman function, that $\rho(u) \approx u^{-u}$ for large $u$,
is shown to be universal for this class.
  A special case of our bounds arises when $X$ is a sum of independent random
variables, each admitting a 1-bounded size bias coupling. In this case, our
bounds are comparable to Chernoff--Hoeffding bounds; however, ours are broader
in scope, sharper for the upper tail, and equal for the lower tail.
  We discuss \emph{bounded} and \emph{monotone} couplings, give a sandwich
principle, and show how this gives an easy conceptual proof that any finite
positive mean sum of independent Bernoulli random variables admits a 1-bounded
coupling with the same conditioned to be nonzero.
