发表机构
Shandong University（山东大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为典范过程研究中的距离族证明了不依赖分布的确定性结果，包括从增长条件构造可容许划分，以及用参数化分离树表示并与控制测度量比较，并推广至伯努利过程。
AI 中文摘要
我们证明了在典范过程研究中出现的距离族的两类确定性结果。第一类结果从增长条件推导出一个可容许划分方案。第二类结果给出了用参数化分离树表示的形式，并将其与相应的控制测度量进行比较。要点在于证明不依赖于底层过程的分布：一旦给定初始距离和距离族，就不再使用随机变量、独立性、尾部函数或矩估计。对于具有正则对数凹尾部的典范过程，抽象结果的假设可由通常的正则性条件推出。分离树估计的一个方向也适用于伯努利过程，无需这些额外假设，并且我们证明了有界凸无约束指标集的逆向估计。我们还给出了一个针对点的增长论证版本，对于有限指标集，该论证导致可容许划分和参数化分离树的递归构造。
英文摘要
We prove two deterministic results for families of distances arising in the study of canonical processes. The first derives an admissible partition scheme from a growth condition. The second gives a representation in terms of parameterized separation trees and compares it with the corresponding majorizing-measure quantities. The main point is that the proofs do not depend on the distribution of the underlying process: once the initial distance and the family of distances are given, no random variables, independence, tail functions, or moment estimates are used. For canonical processes with regular log-concave tails, the assumptions of the abstract results follow from the usual regularity conditions. One direction of the separation-tree estimate also applies to Bernoulli processes without these additional assumptions, and we prove the reverse estimate for bounded convex unconditional index sets. We also give a version of the growth argument for points which, for finite index sets, leads to a recursive construction of admissible partitions and parameterized separation trees.
