Bounded size bias coupling: a Gamma function bound, and universal Dickman-function behavior

Abstract

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.