Using Sums-of-Squares to Prove Gaussian Product Inequalities

2022/05/04 by Oliver Russell, Wei Sun, Russell, Oliver +1 · 1 citation
Mathematics · Medicine · #Analytic Number Theory Research #FOS: Mathematics #Phytoestrogen effects and research #Primary 60E15 #Probability (math.PR) #Secondary 62H12

paper · pdf · doi:10.48550/arxiv.2205.02127

openalex publication_date 2022/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The long-standing Gaussian product inequality (GPI) conjecture states that E [∏_j=1ⁿX_j^2m_j]≥∏_j=1ⁿE[X_j^2m_j] for any centered Gaussian random vector (X₁,…,X_n) and m₁,…,m_n∈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[X₁^2m₁X₂⁶X₃⁴]≥ E[X₁^2m₁]E[X₂⁶]E[X₃⁴] and E[X₁^2m₁X₂²X₃²X₄²]≥ E[X₁^2m₁]E[X₂²]E[X₃²]E[X₄²].

Cited by

• Results related to the Gaussian product inequality conjecture for mixed-sign exponents in arbitrary dimension

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