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.
From: Lasse Leskelä [view email]
[v1]
Sat, 16 Aug 2025 16:54:13 UTC (364 KB)
[v2]
Tue, 23 Jun 2026 11:32:21 UTC (372 KB)
[v3]
Mon, 3 Aug 2026 07:54:26 UTC (372 KB)
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