On positive association of absolute-valued and squared multivariate Gaussians beyond MTP2

We show that positively associated squared (and absolute-valued) multivariate normally distributed random vectors need not be multivariate totally positive of order 2 (MTP₂) for $p\ge 3$. This result disproves Theorem 1 in Eisenbaum (2014, Ann. Probab.) and the conjecture that positive association of squared multivariate normals is equivalent to MTP₂ and infinite divisibility of squared multivariate normals. Among others, we show that there exist absolute-valued multivariate normals which are conditionally increasing in sequence (CIS) (or weakly CIS (WCIS)) and hence positively associated but not MTP₂. Moreover, we show that there exist absolute-valued multivariate normals which are positively associated but not CIS. As a by-product, we obtain necessary conditions for CIS and WCIS of absolute normals. We illustrate these conditions in some examples. With respect to implications and applications of our results, we show PA beyond MTP₂ for some related multivariate distributions (chi-square, $t$, skew normal) and refer to possible conservative multiple test procedures and conservative simultaneous confidence bounds. Finally, we obtain the validity of the strong form of Gaussian product inequalities beyond MTP₂.