Research Article · Preprint Mathematics
Local and eventual entropy concavity under Gaussian smoothing
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics · Classical and Real Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
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. A common all-weight threshold for the full symmetric log-concave class remains open.
Keywords Probability and Statistics · Classical and Real Analysis · mathematical research
Scope of the work
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; no curvature counterexample.
Reproduction notes
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A stable scholarly record
Cite this edition
Scott Schubr. (2026). Local and eventual entropy concavity under Gaussian smoothing (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/local-and-eventual-entropy-concavity-under-gaussian-smoothing/v/1
BibTeX
@techreport{luminary_local_and_eventual_entropy_concavity_under_gaussian_smoothing_v1,
title = {Local and eventual entropy concavity under Gaussian smoothing},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/local-and-eventual-entropy-concavity-under-gaussian-smoothing/v/1}
}Loading discussion…