The three-dimensional Gaussian product inequality

Lan, Guanghui; Hu, Ze-Chun; Sun, Wei

doi:10.1016/j.jmaa.2020.123858

Cited by 14 publications

(10 citation statements)

References 13 publications

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“…This conjecture was shown to be true for d = 2 (see Theorem 2.1 in [2]), for d = 3 (see Theorem 1.1 in [5]), and whenever |X| := (|X 1 |, |X 2 |, . .…”

Section: Introductionmentioning

confidence: 95%

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A combinatorial proof of the Gaussian product inequality beyond the MTP${}_2$ case

Genest¹,

Ouimet²

2021

Preprint

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In this paper, we present a combinatorial proof of the Gaussian product inequality (GPI) conjecture in all dimensions when the components of the centered Gaussian vector X = (X 1 , X 2 , . . . , X d ) can be written as linear combinations, with nonnegative coefficients, of the components of a standard Gaussian vector. The proof comes down to the monotonicity of a certain ratio of gamma functions. We also show that our condition is weaker than assuming the vector of absolute values |X| := (|X 1 |, |X 2 |, . . . , |X d |) to be in the multivariate totally positive of order 2 (MTP 2 ) class on [0, ∞) d , for which the conjecture is already known to be true.

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“…This conjecture was shown to be true for d = 2 (see Theorem 2.1 in [2]), for d = 3 (see Theorem 1.1 in [5]), and whenever |X| := (|X 1 |, |X 2 |, . .…”

Section: Introductionmentioning

confidence: 95%

“…The Gaussian product inequality (GPI) conjecture has gained traction lately following the proof of the three-dimensional case by Lan et al [5] and the proof of the closely related Gaussian correlation inequality by Royen [11] (in fact, Royen proved an extension involving Gamma random variables), see also Lata la & Matlak [6].…”

Section: Introductionmentioning

confidence: 99%

A combinatorial proof of the Gaussian product inequality beyond the MTP${}_2$ case

Genest¹,

Ouimet²

2021

Preprint

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“…The weak form of the GPI was established by Frenkel [4] for n 1 = • • • = n d = 2 and arbitrary d, and by Lan, et al [11] for d = 3 and integer exponents n 1 , n 2 , and n 3 with equality between at least two exponents. Genest and Ouimet [5] developed recently a novel and far-reaching approach to the GPI, proving (1.2) for arbitrary d with nonnegative even integer exponents n j when the covariance matrix Σ is completely positive, i.e., Σ = CC ′ where C = (c ij ) is a d×d matrix with c ij ≥ 0 for all i, j = 1, .…”

Section: Introductionmentioning

confidence: 97%

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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“…In [4,Theorem 3.2], Lan et al used the Gaussian hypergeometric functions to prove the following inequality: for any m 1 , m 2 ∈ N and any centered Gaussian random vector (X 1 , X 2 , X 3 ),…”

Section: Introductionmentioning

confidence: 99%

“…Note that the assumption X 2 and X 3 have the same exponent m 2 plays an essential role in the proof of [4,Theorem 3.2]. A natural question is whether the three-dimensional GPI still holds when the exponents of X 1 , X 2 , X 3 are all different.…”

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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The three-dimensional Gaussian product inequality

Cited by 14 publications

References 13 publications

A combinatorial proof of the Gaussian product inequality beyond the MTP${}_2$ case

A combinatorial proof of the Gaussian product inequality beyond the MTP${}_2$ case

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

Some New Gaussian Product Inequalities

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