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

TL;DR

This paper develops a nonasymptotic framework for sub-Poisson distributions, introducing an optimal variance proxy, deriving concentration inequalities, and establishing closure properties, thereby unifying the treatment of Bernoulli and Poisson variables.

Contribution

It introduces a new notion of sub-Poisson variance proxy and derives Bennett-type inequalities without boundedness assumptions, expanding the understanding of sub-Poisson distributions.

Findings

Derived Bennett-type concentration inequality for sub-Poisson distributions.

Showed sub-Poisson property is closed under independent sums and convex combinations.

Connected sub-Poisson variance proxy to sub-Gaussian and sub-exponential norms.

Abstract

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.