An Opposite Gaussian Product Inequality

Abstract

The long-standing Gaussian product inequality (GPI) conjecture states thatfor any centered Gaussian random vector (X 1 , . . . , X n ) and any non-negative real numbers α j , j = 1, . . . , n. In this note, we prove a novel "opposite GPI" for centered bivariate Gaussian random variables when −1 < α 1 < 0 and α 2 > 0:This completes the picture of bivariate Gaussian product relations.

Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices

“…This is a special case of the strong Gaussian product inequality conjecture, inequality (1). If the covariance matrix is singular, then an even stronger inequality holds, see Proposition 3.1 (ii) of Russell & Sun [24]. The d-dimensional analog of inequality (10) was shown to hold by Frenkel [6] for…”

“…Russell & Sun [24] recently showed, using a moment formula from Nabeya [18], that the reverse inequality in (1) is valid in dimension d = 2 when (α 1 , α 2 ) ∈ (−1, 0] × [0, ∞), thereby completing the study of the GPI conjecture in the bivariate case. In arbitrary dimension, Russell & Sun [25] also proved inequality (1) for all α 1 , .…”