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

Abstract

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.

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

“…In this article, we derive new and more general hypotheses under which the weak form of the GPI (1.2) and the strong form of the GPI (1.3) hold. 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.…”

“…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.…”

“…Genest and Ouimet [5] established the weak form of the GPI (1.2) for the multivariate normal distribution N d (0, Σ) and with even integers n 1 , . .…”

“…We refer readers to Frenkel [4] and Genest and Ouimet [5] for details of the history, motivation, and literature on this inequality. Several generalizations of (1.1) have been studied recently.…”

“…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, . .…”