2022/01/11 by Oliver Russell, Wei Sun, Russell, Oliver +1
Decision Sciences · Engineering · Mathematics · #60E15 #62H12 #FOS: Mathematics #Guidance and Control Systems #Point processes and geometric inequalities #Probability (math.PR) #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.2201.04242
openalex publication_date 2022/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Gaussian product inequality is a long-standing conjecture. In this paper, we investigate the three-dimensional inequality E[X12X22m2X32m3]≥ E[X12]E[X22m2]E[X32m3] for any centered Gaussian random vector (X1,X2,X3) and m2,m3∈ℕ. First, we show that this inequality is implied by a combinatorial inequality. The combinatorial inequality can be verified directly for small values of m2 and arbitrary m3. Hence the corresponding cases of the three-dimensional inequality are proved. Second, we show that the three-dimensional inequality is equivalent to an improved Cauchy-Schwarz inequality. This observation leads us to derive some novel moment inequalities for bivariate Gaussian random variables.