On positive association of absolute-valued and squared multivariate Gaussians beyond MTP2
本文证明对于p≥3，平方（及绝对值）多元正态随机向量的正关联性并不需要满足MTP2性质，推翻了Eisenbaum（2014）的定理及相关猜想，并给出了条件递增序列（CIS）等性质的必要条件。
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≥3. This result disproves Theorem 1 in Eisenbaum (2014, Ann. Prob.) 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.