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The Gaussian product inequality conjecture for multinomial covariances

3 September 2022
Frédéric Ouimet
ArXiv (abs)PDFHTML
Abstract

In this paper, we find an equivalent combinatorial condition only involving finite sums (which is appealing from a numerical point of view) under which the centered Gaussian random vector with multinomial covariance, (X1,X2,…,Xd)∼Nd(0d,diag(p)−pp⊤)(X_1,X_2,\dots,X_d) \sim \mathcal{N}_d(\boldsymbol{0}_d, \mathrm{diag}(\boldsymbol{p}) - \boldsymbol{p} \boldsymbol{p}^{\top})(X1​,X2​,…,Xd​)∼Nd​(0d​,diag(p)−pp⊤), satisfies the Gaussian product inequality (GPI), namely \mathbb{E}\left[\prod_{i=1}^d X_i^{2m}\right] \geq \prod_{i=1}^d \mathbb{E}\left[X_i^{2m}\right], \quad m\in \mathbb{N}. These covariance matrices are relevant since their off-diagonal elements are negative, which is the hardest case to cover for the GPI conjecture, as mentioned by Russell & Sun (2022). Numerical computations provide evidence for the validity of the combinatorial condition.

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