Using Sums-of-Squares to Prove Gaussian Product Inequalities

Russell, Oliver; Sun, Wei

doi:10.48550/arxiv.2205.02127

Cited by 4 publications

(5 citation statements)

References 10 publications

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“…Another extension, valid in dimension d = 3 and for all (α 1 , α 2 , α 3 ) ∈ {1}×{2, 3}×N 0 was recently obtained by Russell & Sun [25] using a brute-force combinatorial approach. Finally, using a sums-of-squares approach along with extensive symbolic/numerical computations in Macaulay2 and Mathematica, Russell & Sun [26] recently stated in their Theorems 3.1 and 3.2 the validity of inequality ( 14) when (d, α 1 , α 2 , α 3 ) = (3, 4, 3, 2) and (d, α 1 , α 2 , α 3 , α 4 ) = (4, 2, 1, 1, 1), respectively.…”

Section: Closing Commentsmentioning

confidence: 99%

Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices

Genest¹,

Ouimet²

2022

Preprint

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This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei [32] and Liu et al. [15] are extended in two ways: by replacing the power functions with more general classes of functions and by replacing the usual Gaussian and multivariate gamma distributional assumptions by the more general trace-Wishart distribution assumption. These findings suggest that a Kronecker product form of the GPI holds for diagonal blocks of any Wishart distribution.

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Section: Closing Commentsmentioning

confidence: 99%

Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices

Genest¹,

Ouimet²

2022

Preprint

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“…Edelmann et al [5] extended (1.7) to the multivarite gamma distributions. As to other related work, we refer to Genest and Ouiment [8], Russell and Sun [18], and Russell and Sun [19].…”

Section: Introductionmentioning

confidence: 99%

Quantitative versions of the two-dimensional Gaussian product inequalities

Han

Zhou

2023

J Inequal Appl

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The Gaussian product-inequality (GPI) conjecture is one of the most famous inequalities associated with Gaussian distributions and has attracted much attention. In this note, we investigate the quantitative versions of the two-dimensional Gaussian product inequalities. For any centered, nondegenerate, and two-dimensional Gaussian random vector $(X_{1}, X_{2})$ ( X 1 , X 2 ) with $E[X_{1}^{2}]=E[X_{2}^{2}]=1$ E [ X 1 2 ] = E [ X 2 2 ] = 1 and the correlation coefficient ρ, we prove that for any real numbers $\alpha _{1}, \alpha _{2}\in (-1,0)$ α 1 , α 2 ∈ ( − 1 , 0 ) or $\alpha _{1}, \alpha _{2}\in (0,\infty )$ α 1 , α 2 ∈ ( 0 , ∞ ) , it holds that $$ {\mathbf{E}}\bigl[ \vert X_{1} \vert ^{\alpha _{1}} \vert X_{2} \vert ^{\alpha _{2}}\bigr]-{\mathbf{E}}\bigl[ \vert X_{1} \vert ^{ \alpha _{1}}\bigr]{\mathbf{E}}\bigl[ \vert X_{2} \vert ^{\alpha _{2}}\bigr]\ge f(\alpha _{1}, \alpha _{2}, \rho )\ge 0, $$ E [ | X 1 | α 1 | X 2 | α 2 ] − E [ | X 1 | α 1 ] E [ | X 2 | α 2 ] ≥ f ( α 1 , α 2 , ρ ) ≥ 0 , where the function $f(\alpha _{1}, \alpha _{2}, \rho )$ f ( α 1 , α 2 , ρ ) will be given explicitly by the Gamma function and is positive when $\rho \neq 0$ ρ ≠ 0 . When $-1<\alpha _{1}<0$ − 1 < α 1 < 0 and $\alpha _{2}>0$ α 2 > 0 , Russell and Sun (Statist. Probab. Lett. 191:109656, 2022) proved the “opposite Gaussian product inequality”, of which we will also give a quantitative version. These quantitative inequalities are derived by employing the hypergeometric functions and the generalized hypergeometric functions.

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“…(g) For all (n 1 , n 2 , n 3 ) ∈ N 0 × {3} × {2}, GPI 3 (n 1 , n 2 , n 3 ) holds, as shown by Theorem 4.1 of Russell & Sun [30], taking advantage of a sums-of-squares methodology combined with in-depth symbolic/numerical calculations using Macaulay2 and Mathematica.…”

Section: Introductionmentioning

confidence: 99%

“…(h) For all (n 1 , n 2 , n 3 , n 4 ) ∈ N 0 × {2} × {2} × {2}, GPI 4 (n 1 , n 2 , n 3 , n 4 ) holds, as shown by Theorem 4.2 of Russell & Sun [30], using the same methodology as for (g).…”

Section: Introductionmentioning

confidence: 99%