Using Sums-of-Squares to Prove Gaussian Product Inequalities

Oliver Russell; Wei Sun

paper

Using Sums-of-Squares to Prove Gaussian Product Inequalities

arXiv:2205.02127

Abstract

The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j = 1}^{n} X_{j}^{2 m_{j}}] \geq \prod_{j = 1}^{n} E [X_{j}^{2 m_{j}}]$ for any centered Gaussian random vector $(X_{1}, \dots, X_{n})$ and $m_{1}, \dots, m_{n} \in N$ . In this paper, we describe a computational algorithm involving sums-of-squares representations of multivariate polynomials that can be used to resolve the GPI conjecture. To exhibit the power of this novel method, we apply it to prove two new GPIs: $E [X_{1}^{2 m_{1}} X_{2}^{6} X_{3}^{4}] \geq E [X_{1}^{2 m_{1}}] E [X_{2}^{6}] E [X_{3}^{4}]$ and $E [X_{1}^{2 m_{1}} X_{2}^{2} X_{3}^{2} X_{4}^{2}] \geq E [X_{1}^{2 m_{1}}] E [X_{2}^{2}] E [X_{3}^{2}] E [X_{4}^{2}]$ .

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