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

Lasse Leskelä, Ian Välimaa

Sankhya A | Jul 16, 2026

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.

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Authors

Researchers on this paper

Lasse Leskelä

first | Aalto University | ORCID 0000-0001-8411-8329

Ian Välimaa

last | Aalto University

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Citation

BibTeX

@article{Leskel2026Poisson,
  title = {Sub-Poisson Distributions: Concentration Inequalities, Optimal Variance Proxies, and Closure Properties},
  author = {Lasse Leskelä and Ian Välimaa},
  journal = {Sankhya A},
  year = {2026},
  doi = {10.1007/s13171-026-00444-x},
  url = {https://doi.org/10.1007/s13171-026-00444-x}
}

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