Local and eventual entropy concavity under Gaussian smoothing
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Abstract
For independent copies of a symmetric variance-one input, we study entropy of their variance-preserving weighted sum after Gaussian smoothing. For every Q < 1/2, we prove all-weight strict entropy concavity for 0 < q <= Q in an explicit infinite-dimensional Gaussian-relative bounded-density neighborhood. A uniform moment envelope with finite nonzero Hermite discrepancy gives a common eventual threshold; every fixed symmetric non-Gaussian all-moment input has eventual all-weight strict concavity. For the symmetric log-concave class, any counterexample sequence as q decreases to zero must converge weakly and in each fixed absolute moment to the Gaussian, while endpoint curvature is strictly negative for q <= 1/20. An exact moment-matched exponential-tail construction delineates the remaining near-Gaussian gap.
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Research details
- Subjects
- Mathematics · Probability and Statistics · Classical and Real Analysis
- Keywords
- Probability and Statistics, Classical and Real Analysis, mathematical research
- Manuscript date
- 2026-10-23
- Reader reuse
- No additional permission granted
Scope
Explicit all-weight relative-density neighborhood for every fixed Q<1/2 and every0<q<=Q. Finite Hermite-discrepancy uniform restoration and Gaussian-only weak/moment limiting obstruction. Per-input all-moments eventual restoration; whole logconcave-class endpoints q<=1/20. Moment-matched exponential tails distinguish the remaining nearGaussian topology.
Disclosures
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