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Product inequalities for multivariate Gaussian, gamma, and positively upper orthant dependent distributions

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  • Edelmann, Dominic
  • Richards, Donald
  • Royen, Thomas

Abstract

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 (2021). 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.

Suggested Citation

  • Edelmann, Dominic & Richards, Donald & Royen, Thomas, 2023. "Product inequalities for multivariate Gaussian, gamma, and positively upper orthant dependent distributions," Statistics & Probability Letters, Elsevier, vol. 197(C).
    • Handle: RePEc:eee:stapro:v:197:y:2023:i:c:s0167715223000445
    DOI: 10.1016/j.spl.2023.109820
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    References listed on IDEAS

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    Cited by:

    1. Finner, Helmut & Roters, Markus, 2024. "On positive association of absolute-valued and squared multivariate Gaussians beyond MTP2," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    2. Russell Oliver & Sun Wei, 2024. "Using sums-of-squares to prove Gaussian product inequalities," Dependence Modeling, De Gruyter, vol. 12(1), pages 1-13.

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