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Gaussian product inequality conjecture

Let X=(X1,…,Xn)\boldsymbol{X} = (X_1,\ldots,X_n) be a centered Gaussian vector, not necessarily nondegenerate. Then, for every α1,…,αn>0\alpha_1,\ldots,\alpha_n > 0, E[∏i=1n∣Xi∣αi]≥∏i=1nE[∣Xi∣αi].\mathsf{E}\left[\prod_{i=1}^n |X_i|^{\alpha_i}\right] \geq \prod_{i=1}^n \mathsf{E}\left[|X_i|^{\alpha_i}\right]. Moreover, if Var(Xi)>0\mathsf{Var}(X_i) > 0 for every ii, then equality holds if and only if X1,…,XnX_1,\ldots,X_n are independent.

Result
Proved
Status
Resolved
AI contribution
AI-discovered
Method
Argument
Field
Probability & statistics
Posed by
Péter E. Frenkel
Year posed
2007
Years open
19y
Solved
2026-07-20
Model
ChatGPT 5.6 Sol
Vendor
—
Collaborators
Frédéric Ouimet, Dylan Greaves
Verification
Lean-verified
Publication
Announced
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What the AI did

The AI provided a complete and correct solution without the characterization of equality in terms of independence (but only because the equality case was not in the original prompt by Dylan Greaves).

Verification

The prompt and output are available at https://chatgpt.com/share/6a5ea69b-1648-83e8-80b1-014ae0b1003c. This early version of the proof was formalized in Lean using Codex; see https://github.com/dylgre/gaussian-product-inequality. The proof has also been checked by ChatGPT 5.6 Sol (Pro), Gemini 3.1 Pro (Extended Thinking), Grok 4.5 (Expert), and Frédéric Ouimet.

Source

A proof of the strong Gaussian product inequality conjecture

Submitted by JollyJackal127

Changelog6 changes
  • JollyJackal127changed Statement from “Let $\boldsymbol{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily non…” to “Let $\boldsymbol{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily non…”
  • JollyJackal127changed Statement from “Let $\bb{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily nondegenera…” to “Let $\boldsymbol{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily non…”
  • JollyJackal127set Citation URL to “https://doi.org/10.13140/RG.2.2.17569.77923/1”
  • JollyJackal127set Citation source to “Research Gate”
  • Rasmus Lindahlapproved this entry
  • JollyJackal127submitted this entry

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