Variance and local log-concavity of Poisson–binomial laws
Year: 2026
Type: preprint
Abstract: Let W be the number of successes in independent trials with unequal success probabilities, with probability mass function f and variance V ≥ 1, and let D be the first index at which f decreases. The normalized Turán deficit δ_k = 1 − f_{k−1}f_{k+1}/f_k², a measure of local log-concavity, is a bounded transform of the discrete curvature of log f at k; for a normal density of variance V, that curvature is 1/V. Quantitative log-concavity is used to derive concentration and log-Sobolev inequalities for discrete laws, but for Bernoulli sums the known lower bounds on δ_k depend on the number of summands or on the success probabilities. We prove V·δ_D ≥ 1/4, and binomial laws show that no constant above 1/3 is possible. With cubic inequalities of Hillion and Johnson, this gives δ_k ≥ 1/(4V + |k − D|) on the whole support. Ultra-log-concavity alone gives no variance-scaled lower bound on δ_D. For the number of eigenvalues of a Haar unitary matrix in a half circle, our bound at D is of order 1/log N, against order 1/N from ultra-log-concavity. The proof of the main theorem is computer-assisted and is verified in the Lean proof assistant. Preprint, submitted to Electronic Communications in Probability (October 2026). MSC2020: 60E15 (primary); 60C05, 05A20 (secondary). Code, exact certificate data and the Lean formalizations: https://doi.org/10.5281/zenodo.23129480.
Author: Reynolds, Brett
Institutions: Humber Polytechnic, University of Toronto
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7219565778
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