Using sums-of-squares to prove Gaussian product inequalities

Abstract

The long-standing Gaussian product inequality (GPI) conjecture states that E[∏j=1n∣Xj∣yj]≥∏j=1nE[∣Xj∣yj]E\left[{\prod }_{j=1}^{n}{| {X}_{j}| }^{{y}_{j}}]\ge {\prod }_{j=1}^{n}E\left[{| {X}_{j}| }^{{y}_{j}}] for any centered Gaussian random vector (X1,…,Xn)\left({X}_{1},\ldots ,{X}_{n}) and any non-negative real numbers yj{y}_{j}, j=1,…,nj=1,\ldots ,n. In this study, 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 the novel method, we apply it to prove new four- and five-dimensional GPIs: E[X12mX22X32X42]≥E[X12m]E[X22]E[X32]E[X42]E\left[{X}_{1}^{2m}{X}_{2}^{2}{X}_{3}^{2}{X}_{4}^{2}]\ge E\left[{X}_{1}^{2m}]E\left[{X}_{2}^{2}]E\left[{X}_{3}^{2}]E\left[{X}_{4}^{2}] for any m∈Nm\in {\mathbb{N}}, and E[∣X1∣yX22X32X42X52]≥E[∣X1∣y]E[X22]E[X32]E[X42]E[X52]E\left[{| {X}_{1}| }^{y}{X}_{2}^{2}{X}_{3}^{2}{X}_{4}^{2}{X}_{5}^{2}]\ge E\left[{| {X}_{1}| }^{y}]E\left[{X}_{2}^{2}]E\left[{X}_{3}^{2}]E\left[{X}_{4}^{2}]E\left[{X}_{5}^{2}] for any y≥110y\ge \frac{1}{10}.

Keywords

• moments of gaussian random vector
• gaussian product inequality conjecture
• sums-of-squares
• semi-definite programming
• primary 60e15
• secondary 62h12