Using sums-of-squares to prove Gaussian product inequalities

Russell Oliver () and Sun Wei ()
Additional contact information
Russell Oliver: Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, H3G 1M8, Canada
Sun Wei: Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, H3G 1M8, Canada

Dependence Modeling, 2024, vol. 12, issue 1, 13

Abstract: The long-standing Gaussian product inequality (GPI) conjecture states that E [ ∏ j = 1 n ∣ X j ∣ y j ] ≥ ∏ j = 1 n E [ ∣ X j ∣ y j ] E\left[{\prod }_{j=1}^{n}{| {X}_{j}| }^{{y}_{j}}]\ge {\prod }_{j=1}^{n}E\left[{| {X}_{j}| }^{{y}_{j}}] for any centered Gaussian random vector ( X 1 , … , X n ) \left({X}_{1},\ldots ,{X}_{n}) and any non-negative real numbers y j {y}_{j} , j = 1 , … , n j=1,\ldots ,n . In this study, 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 the novel method, we apply it to prove new four- and five-dimensional GPIs: E [ X 1 2 m X 2 2 X 3 2 X 4 2 ] ≥ E [ X 1 2 m ] E [ X 2 2 ] E [ X 3 2 ] E [ X 4 2 ] E\left[{X}_{1}^{2m}{X}_{2}^{2}{X}_{3}^{2}{X}_{4}^{2}]\ge E\left[{X}_{1}^{2m}]E\left[{X}_{2}^{2}]E\left[{X}_{3}^{2}]E\left[{X}_{4}^{2}] for any m ∈ N m\in {\mathbb{N}} , and E [ ∣ X 1 ∣ y X 2 2 X 3 2 X 4 2 X 5 2 ] ≥ E [ ∣ X 1 ∣ y ] E [ X 2 2 ] E [ X 3 2 ] E [ X 4 2 ] E [ X 5 2 ] E\left[{| {X}_{1}| }^{y}{X}_{2}^{2}{X}_{3}^{2}{X}_{4}^{2}{X}_{5}^{2}]\ge E\left[{| {X}_{1}| }^{y}]E\left[{X}_{2}^{2}]E\left[{X}_{3}^{2}]E\left[{X}_{4}^{2}]E\left[{X}_{5}^{2}] for any y ≥ 1 10 y\ge \frac{1}{10} .

Keywords: moments of Gaussian random vector; Gaussian product inequality conjecture; sums-of-squares; semi-definite programming (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1515/demo-2024-0003 (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:vrs:demode:v:12:y:2024:i:1:p:13:n:1001

DOI: 10.1515/demo-2024-0003

Access Statistics for this article

Dependence Modeling is currently edited by Giovanni Puccetti

More articles in Dependence Modeling from De Gruyter
Bibliographic data for series maintained by Peter Golla ().