Work

Sub-Poisson Distributions: Concentration Inequalities, Optimal Variance Proxies, and Closure Properties

Year: 2026

Type: article

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.

Source: Sankhya A

DOI: 10.1007/s13171-026-00444-x

Authors: Lasse Leskelä, Ian Välimaa

Institution: Aalto University

Language: English

Topic: Probability and Statistical Research

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funders: Aalto-Yliopisto, Emil Aaltosen Säätiö