On the Gaussian Product Inequality

The long-standing Gaussian product inequality (GPI) conjecture states that E[|X_1|^{y_1} |X_2|^{y_2} ··· |X_n|^{y_n}] ≥ E[|X_1|^{y_1}] E[|X_2|^{y_2}] ··· E[|X_n|^{y_n}] for any centered Gaussian random vector (X_1 , . . . , X_n) and any non-negative real numbers y_j , j = 1, . . . , n. First, we complete the picture of bivariate Gaussian product relations by proving a novel “opposite GPI” when -1<y_1<0 and y_2>0: E[|X_1|^{y_1} |X_2|^{y_2}] ≤ E[|X_1|^{y_1}] E[|X_2|^{y_2}]. Next, we investigate the three-dimensional inequality E[X_1^2 X_2^{2m_2} X_n^{2m_3}] ≥ E[X_1^2] E[X_2^{2m_2}] E[X_n^{2m_3}] for any natural numbers m_2, m_3. We show that this inequality is implied by a combinatorial inequality which we verify directly for small values of m_2 and arbitrary m_3. Then, we complete the proof through the discovery of a novel moment ratio inequality which implies this three-dimensional GPI. We then extend these three-dimensional results to the case where the exponents in the GPI can be real numbers rather than simply even integers. Finally, we describe two computational algorithms involving sums-of-squares representations of polynomials that can be used to resolve the GPI conjecture. To exhibit the power of these novel methods, we apply them to prove new four- and five-dimensional GPIs.

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