发表机构
School of Management, University of Science and Technology of China（中国科学技术大学管理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过构造最小状态空间上的精确反例，证明完全负回归相依既不蕴含尾部相依也不蕴含负关联，并解决了相关开放问题。
AI 中文摘要
我们证明完全负回归相依既不蕴含负右尾相依，也不蕴含负左尾相依。在2*2*3矩形状态空间上的一个严格正分布为右尾情形提供了精确反例，其坐标反射为左尾情形提供了反例。我们进一步证明该状态空间大小是最小的。该反例通过使用精确整数运算的有限上集准则得以确立。我们还解决了完全负回归相依是否蕴含负关联这一开放问题。{0,1,2}^4上的一个严格正分布满足完全负回归相依却违反负关联；该反例由17532个精确整数不等式验证。本文的主要结果最初由多智能体数学系统Eureka发现，随后由作者验证。
英文摘要
We show that full negative regression dependence implies neither negative right-tail dependence nor negative left-tail dependence. A strictly positive distribution on a 2*2*3 rectangular state space provides an exact counterexample for the right-tail case, and its coordinatewise reflection yields a counterexample for the left-tail case. We further prove that this state-space size is minimal. The counterexample is established via a finite upper-set criterion using exact integer arithmetic. We also resolve the open question of whether full negative regression dependence implies negative association. A strictly positive distribution on {0,1,2}^4 satisfies full negative regression dependence yet violates negative association; this counterexample is verified by 17532 exact integer inequalities. The main results of this paper were first uncovered by the multi-agent mathematical system Eureka and later verified by the authors.
