Local and eventual mixing-weight entropy concavity under Gaussian smoothing
Loading original first page…
Abstract
For independent copies of a symmetric variance-one input, we study entropy of their variance-preserving weighted sum after Gaussian smoothing with retained signal fraction q. For every Q<1/2, we prove all-weight strict entropy concavity, simultaneously for 0<q≤ Q, in an explicit infinite-dimensional Gaussian-relative L^∞ neighborhood. We also prove a common all-weight threshold for symmetric all-moment families with a uniform moment envelope and a finite nonzero Hermite discrepancy. Consequently, among symmetric variance-one log-concave inputs, any counterexample sequence with q↓0 must converge weakly and in every fixed absolute moment to the Gaussian. Each fixed symmetric non-Gaussian all-moment input has eventual all-weight strict concavity, and the whole log-concave class has strictly negative endpoint curvature for 0<q≤1/20. An exact moment-matching exponential-tail construction delineates the remaining near-Gaussian gap. No common all-weight threshold for the full symmetric log-concave class is established.
Read the full paper in the original PDF.
Edition files
- Paper.pdf131.1 KiB · pdf
SHA-256
25de2d9ac5116c3722b0f116859f80165fe8bec140a3effac4cbe2edba6e5278 - Publication_Package.zip145.5 KiB · package
SHA-256
c297a7d679a33bc84d31ad50beac5eae522873849a177ea77632892a99a91b91
Available editions
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
Local Gaussian-relative and fixed-input eventual all-weight entropy concavity, uniform finite-Hermite-discrepancy families, and a common endpoint-curvature theorem for symmetric log-concave inputs.
Disclosures
AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.
Reproduction
14-page manuscript and 6 package members. Compile main.tex directly or run python build.py --tectonic tectonic using Python 3. The script fixes SOURCE_DATE_EPOCH; Tectonic uses its standard support bundle. The mathematical proof and bibliography are contained in main.tex, with no external numerical dataset.