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An opposite Gaussian product inequality

Author

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  • Russell, Oliver
  • Sun, Wei

Abstract

The long-standing Gaussian product inequality (GPI) conjecture states that E[∏j=1n|Xj|αj]≥∏j=1nE[|Xj|αj] for any centered Gaussian random vector (X1,…,Xn) and any non-negative real numbers αj, j=1,…,n. In this note, we prove a novel “opposite GPI” for centered bivariate Gaussian random variables when −1<α1<0 and α2>0: E[|X1|α1|X2|α2]≤E[|X1|α1]E[|X2|α2]. This completes the picture of bivariate Gaussian product relations.

Suggested Citation

  • Russell, Oliver & Sun, Wei, 2022. "An opposite Gaussian product inequality," Statistics & Probability Letters, Elsevier, vol. 191(C).
    • Handle: RePEc:eee:stapro:v:191:y:2022:i:c:s0167715222001766
    DOI: 10.1016/j.spl.2022.109656
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    References listed on IDEAS

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    1. Wenbo V. Li & Ang Wei, 2012. "A Gaussian Inequality for Expected Absolute Products," Journal of Theoretical Probability, Springer, vol. 25(1), pages 92-99, March.
    2. Ang Wei, 2014. "Representations of the Absolute Value Function and Applications in Gaussian Estimates," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1059-1070, December.
    3. Liu, Zhenxia & Wang, Zhi & Yang, Xiangfeng, 2017. "A Gaussian expectation product inequality," Statistics & Probability Letters, Elsevier, vol. 124(C), pages 1-4.
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    Cited by:

    1. 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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