Sharp support-ratio criterion for variance-proxy sub-Gaussian laws

For a symmetric real law with 0<a≤|X|≤b, which universal ratio b/a guarantees strict sub-Gaussianity with the actual variance: E exp(tX)≤exp(t²E[X²]/2), strictly for t≠0?

Why it matters and what counts as progress

Why it matters: The support criterion would certify concentration without testing a distribution’s full moment-generating function individually.

What counts as progress: A sufficient ratio with the full moment-generating-function proof, and counterexamples beyond it; the sharp target is √3+√2, not merely a fourth-moment threshold.

How the problem developed

15 Oct 2026 · Sharp criterion [1][2][3]
JD Jones, with AI-assisted research and proof development Proved that a symmetric law with 0<a≤|X|≤b is variance-proxy sub-Gaussian whenever b/a≤√3+√2, and gave four-point counterexamples for every larger ratio.

AI contribution

AI-assisted sharp support criterion for variance-proxy sub-Gaussian laws

The support-criterion project reports that any symmetric real probability law with 0<a≤|X|≤b almost surely and b/a≤√3+√2 satisfies E exp(tX)≤exp(t² E[X²]/2). The inequality is strict for nonzero t, the variance proxy is optimal, and every larger universal support ratio admits a symmetric four-point counterexample.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (Lean)

Human understandingFew