An Opposite Gaussian Product Inequality

2022/05/20 by Oliver Russell, Wei Sun, Russell, Oliver +1
Decision Sciences · #60E15 #62H12 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models

paper · pdf · doi:10.48550/arxiv.2205.10231

openalex publication_date 2022/05/20 · 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=1ⁿ|X_j|^α_j]≥∏_j=1ⁿE[|X_j|^α_j] for any centered Gaussian random vector (X₁,…,X_n) and any non-negative real numbers α_j, j=1,…,n. In this note, we prove a novel "opposite GPI" for centered bivariate Gaussian random variables when -10: E[|X₁|^α₁|X₂|^α₂]≤ E[|X₁|^α₁]E[|X₂|^α₂]. This completes the picture of bivariate Gaussian product relations.

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