Gaussian product inequality conjecture

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probability-statisticsJul 20, 2026Significance 20/100Registry: lean verified

Gaussian product inequality conjecture

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Let $\boldsymbol{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily nondegenerate. Then, for every $\alpha_1,\ldots,\alpha_n > 0$, $$\mathsf{E}\left[\prod_{i=1}^n |X_i|^{\alpha_i}\right] \geq \prod_{i=1}^n \mathsf{E}\left[|X_i|^{\alpha_i}\right].$$ Moreover, if $\mathsf{Var}(X_i) > 0$ for every $i$, then equality holds if and only if $X_1,\ldots,X_n$ are independent.

Source authenticatedGaussian product inequality conjecture →

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Let $\boldsymbol{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily nondegenerate. Then, for every $\alpha_1,\ldots,\alpha_n > 0$, $$\mathsf{E}\left[\prod_{i=1}^n |X_i|^{\alpha_i}\right] \geq \prod_{i=1}^n \mathsf{E}\left[|X_i|^{\alpha_i}\right].$$ Moreover, if $\mathsf{Var}(X_i) > 0$ for every $i$, then equality holds if and only if $X_1,\ldots,X_n$ are independent.

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