Sub-Poisson Distributions: Concentration Inequalities, Optimal Variance Proxies, and Closure Properties
Abstract
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.
Fields of Study & Subjects
Bayesian Methods and Mixture Models
Statistical Methods and Inference
Publication details
- Journal / source
- Sankhya A
- Publisher
- Springer Science and Business Media LLC
- Volume
- 88
- Issue
- 2
- Pages
- 638-657
- Publication date
- 16 July 2026
- Language
- EN