Sub-Poisson distributions: Concentration inequalities, optimal variance proxies, and closure properties

We introduce a nonasymptotic framework for sub-Poisson distributions with moment generating function dominated by that of a Poisson distribution.

At its core is a new notion of optimal sub-Poisson variance proxy, analogous to the variance parameter in the sub-Gaussian setting.

This framework allows us to derive a Bennett-type concentration inequality without boundedness assumptions and to show that the sub-Poisson property is closed under key operations including independent sums and convex combinations, but not under all linear operations such as scalar multiplication.

We derive bounds relating the sub-Poisson variance proxy to sub-Gaussian and sub-exponential Orlicz norms.

Taken together, these results unify the treatment of Bernoulli and Poisson random variables and their signed versions in their natural tail regime.