Representations of the Absolute Value Function and Applications in Gaussian Estimates

Wei, Ang

doi:10.1007/s10959-013-0486-z

Cited by 16 publications

(15 citation statements)

References 13 publications

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“…The strong form of the GPI (1.3) for the case in which all exponents are negative was proved by Wei [20]. We now derive this result succinctly by an application of the Gaussian correlation inequality [14] and the method of integrating the multivariate survival function, as applied earlier in Section 3.…”

Section: The Strong Form Of the Gpi For Negative Exponentsmentioning

confidence: 84%

“…We extend the results of Genest and Ouimet [5] in several directions, one of which is a proof of the strong form of the GPI (1.3) for nonnegative correlations, i.e., for any covariance matrix Σ = (σ ij ) with σ ij ≥ 0 for all 1 ≤ i < j ≤ d. Additionally, we show that the weak form of the GPI and the strong form of the GPI follow from the properties of positive upper orthant dependence (PUOD) and strongly positive upper orthant dependence (SPUOD), respectively. Finally, we apply the Gaussian correlation inequality (Royen [14]) to obtain in Section 4 an alternative and succinct proof of the strong form of the GPI for negative exponents, derived originally by Wei [20]; further, we show that this result extends to the multivariate gamma distributions.…”

Section: Introductionmentioning

confidence: 92%

“…Wei [20] showed, however, that the strong form of the GPI holds for the case in which all exponents n 1 , . .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Product Inequalities for Multivariate Gaussian, Gamma, and Positively Upper Orthant Dependent Distributions

Edelmann¹,

Richards²,

Royen³

2022

Preprint

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The Gaussian product inequality is an important conjecture concerning the moments of Gaussian random vectors. While all attempts to prove the Gaussian product inequality in full generality have been unsuccessful to date, numerous partial results have been derived in recent decades and we provide here further results on the problem. Most importantly, we establish a strong version of the Gaussian product inequality for multivariate gamma distributions in the case of nonnegative correlations, thereby extending a result recently derived by Genest and Ouimet [5]. Further, we show that the Gaussian product inequality holds with nonnegative exponents for all random vectors with positive components whenever the underlying vector is positively upper orthant dependent. Finally, we show that the Gaussian product inequality with negative exponents follows directly from the Gaussian correlation inequality.

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Section: The Strong Form Of the Gpi For Negative Exponentsmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 92%

See 1 more Smart Citation

Product Inequalities for Multivariate Gaussian, Gamma, and Positively Upper Orthant Dependent Distributions

Edelmann¹,

Richards²,

Royen³

2022

Preprint

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show abstract

“…In [7], Frenkel used algebraic methods to prove (1.1) for the case m = 1 (or (1.2) for the case α j = 2) and then used the obtained inequality to improve the lower bound of the 'real linear polarization constant' problem. In [18], Wei used integral representations to prove a stronger version of (1.2) for α j ∈ (−1, 0) as follows.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The three-dimensional Gaussian product inequality

Lan

Sun

2020

Journal of Mathematical Analysis and Applications

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We prove the 3-dimensional Gaussian product inequality, i.e., for any real-valued centered Gaussian random vector (X, Y, Z) and m ∈ N, it holds thatOur proof is based on some improved inequalities on multi-term products involving 2-dimensional Gaussian random vectors. The improved inequalities are derived using the Gaussian hypergeometric functions and have independent interest. As by-products, several new combinatorial identities and inequalities are obtained. MSC: Primary 60E15; Secondary 62H12

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“…In [3], Frenkel used algebraic methods to prove (1.2) for the case that m = 1. In [11], Wei used integral representations to prove a stronger version of (1.1) for α j ∈ (−1, 0) as follows:…”

Section: Introductionmentioning

confidence: 99%

Some New Gaussian Product Inequalities

Russell¹,

Sun²

2022

Preprint

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The Gaussian product inequality is a long-standing conjecture. In this paper, we investigate the three-dimensional inequality E] for any centered Gaussian random vector (X 1 , X 2 , X 3 ) and m 2 , m 3 ∈ N. First, we show that this inequality is implied by a combinatorial inequality. The combinatorial inequality can be verified directly for small values of m 2 and arbitrary m 3 . Hence the corresponding case of the three-dimensional inequality is 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.

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Representations of the Absolute Value Function and Applications in Gaussian Estimates

Cited by 16 publications

References 13 publications

Product Inequalities for Multivariate Gaussian, Gamma, and Positively Upper Orthant Dependent Distributions

Product Inequalities for Multivariate Gaussian, Gamma, and Positively Upper Orthant Dependent Distributions

The three-dimensional Gaussian product inequality

Some New Gaussian Product Inequalities

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