The Gaussian product inequality conjecture for multinomial covariances
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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, (X_1,X_2,…,X_d) ∼𝒩_d(0_d, diag(p) - pp^⊤), satisfies the Gaussian product inequality (GPI), namely 𝔼[∏_i=1^d X_i^2m] ≥∏_i=1^d 𝔼[X_i^2m], m∈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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