无条件对数凹向量在负关联幅值下的Sudakov下界原理
Sudakov minoration for unconditional log-concave vectors with negatively associated magnitudes
- University of Warsaw（华沙大学）
- Faculty of Mathematics, Informatics and Mechanics（数学、信息学和机械学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
证明无条件对数凹随机向量在坐标幅值负关联下具有普适常数的Sudakov下界原理，核心是确定性选择定理结合前缀计数不等式与凸分离，推论包括Orlicz对数凹测度乘积下界及极Lp质心体几何覆盖估计。
AI中文摘要：
我们证明了无条件对数凹随机向量在坐标幅值负关联条件下的Sudakov下界原理，且常数是普适的。主要步骤是一个确定性选择定理：在凸向下闭集合中，两两见证者能在一个指数大的子族上，为每个标签产生一个固定见证者，使其与所有其他保留标签分离。该选择结合了前缀计数不等式与所有标签-坐标数组空间中的凸分离。负关联性随后控制活跃联合尾部见证者的期望数量，而关于互补坐标集的伯努利论证则保留随机符号。证明使用了联合上象限概率，并仅在原测度下应用负关联性。推论包括基于Orlicz的对数凹测度乘积的下界估计，以及在矩度量下的覆盖估计，后者通过极Lp质心体以几何方式表达。
英文摘要：
We prove the Sudakov minoration principle, with a universal constant, for unconditional log-concave random vectors whose coordinate magnitudes are negatively associated. The main step is a deterministic selection theorem: pairwise witnesses in convex downwardclosed sets yield, on an exponentially large subfamily, one fixed witness per label separating it from every other retained label. The selection combines a prefix counting inequality with convex separation in the space of all label-coordinate arrays. Negative association then controls the expected number of active joint-tail witnesses, while a Bernoulli argument on complementary coordinate sets retains the random signs. The proof uses joint upper-orthant probabilities and applies negative association only under the original law. Consequences include minoration for products of Orlicz-based log-concave measures and a covering estimate in the moment metric, expressed geometrically through polar Lp centroid bodies.
