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A combinatorial proof of the Gaussian product inequality beyond the MTP2 case

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  • Genest Christian

    (Department of Mathematics and Statistics, McGill University, 805, rue Sherbrooke ouest, Montréal (Québec), Canada H3A 0B9)

  • Ouimet Frédéric

    (Department of Mathematics and Statistics, McGill University, 805, rue Sherbrooke ouest, Montréal (Québec), Canada H3A 0B9)

Abstract

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector X = ( X 1 , … , X d ) {\boldsymbol{X}}=\left({X}_{1},\ldots ,{X}_{d}) of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on X {\boldsymbol{X}} is shown to be strictly weaker than the assumption that the density of the random vector ( ∣ X 1 ∣ , … , ∣ X d ∣ ) \left(| {X}_{1}| ,\ldots ,| {X}_{d}| ) is multivariate totally positive of order 2, abbreviated MTP 2 {\text{MTP}}_{2} , for which the GPI is already known to hold. Under this condition, the paper highlights a new link between the GPI and the monotonicity of a certain ratio of gamma functions.

Suggested Citation

  • Genest Christian & Ouimet Frédéric, 2022. "A combinatorial proof of the Gaussian product inequality beyond the MTP2 case," Dependence Modeling, De Gruyter, vol. 10(1), pages 236-244, January.
    • Handle: RePEc:vrs:demode:v:10:y:2022:i:1:p:236-244:n:10
    DOI: 10.1515/demo-2022-0116
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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.
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