Author:
Hu Ze-Chun,Zhao Han,Zhou Qian-Qian
Abstract
AbstractThe 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$(X_{1}, X_{2})$(X1,X2)with$E[X_{1}^{2}]=E[X_{2}^{2}]=1$E[X12]=E[X22]=1and the correlation coefficientρ, we prove that for any real numbers$\alpha _{1}, \alpha _{2}\in (-1,0)$α1,α2∈(−1,0)or$\alpha _{1}, \alpha _{2}\in (0,\infty )$α1,α2∈(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[|X1|α1|X2|α2]−E[|X1|α1]E[|X2|α2]≥f(α1,α2,ρ)≥0,where the function$f(\alpha _{1}, \alpha _{2}, \rho )$f(α1,α2,ρ)will be given explicitly by the Gamma function and is positive when$\rho \neq 0$ρ≠0. When$-1<\alpha _{1}<0$−1<α1<0and$\alpha _{2}>0$α2>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.
Funder
National Natural Science Foundation of China
Science Development Project of Sichuan University
Scientific Founda- tion of Nanjing University of Posts and Telecommunications
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis
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