阿贝尔度量群的 Khinchin-Kahane 与 Levy 不等式，以及从赋范（阿贝尔半）群到 Banach 空间的转移
The Khinchin-Kahane and Levy inequalities for abelian metric groups, and transfer from normed (abelian semi)groups to Banach spaces
AI总结：
本文将 Khinchin-Kahane 不等式推广至阿贝尔度量群，提出转移原理将赋范群与半群问题转化为 Banach 空间框架，扩展了期望概念，并构建了涵盖已知变体的广义 Levy 不等式。
摘要：
Khinchin-Kahane 不等式是概率文献中的基本结果，其迄今最一般的版本在 Banach 空间中成立。受现代背景与应用的启发，我们将其推广至任意阿贝尔度量群。若假设群的度量是一个范数（即 $\mathbb{Z}_{>0}$-齐次的），我们解释了该不等式如何改进为与 Banach 空间中相同的形式。这通过一个“转移原理”实现，该原理有助于将涉及赋范度量群与阿贝尔赋范半群的问题转移到 Banach 空间框架中。该原理还将期望的概念扩展到取值于任意阿贝尔赋范度量半群 $\mathscr{G}$ 的随机变量。我们提供了额外应用，包括研究弱 $\ell_p$ $\mathscr{G}$-值序列及相关的 Rademacher 级数。在相关方面，我们还阐述了一个“广义”Levy 不等式，具有两个特征：(i) 它包含了 Banach 空间文献中的多个已知变体；(ii) 我们在表述它所需的最小框架：阿贝尔度量群中证明了该不等式。
英文摘要：
The Khinchin-Kahane inequality is a fundamental result in the probability literature, with the most general version to date holding in Banach spaces. Motivated by modern settings and applications, we generalize this inequality to arbitrary metric groups which are abelian. If instead of abelian one assumes the group's metric to be a norm (i.e., $\mathbb{Z}_{>0}$-homogeneous), then we explain how the inequality improves to the same one as in Banach spaces. This occurs via a "transfer principle" that helps carry over questions involving normed metric groups and abelian normed semigroups into the Banach space framework. This principle also extends the notion of the expectation to random variables with values in arbitrary abelian normed metric semigroups $\mathscr{G}$. We provide additional applications, including studying weakly $\ell_p$ $\mathscr{G}$-valued sequences and related Rademacher series. On a related note, we also formulate a "general" Levy inequality, with two features: (i) It subsumes several known variants in the Banach space literature; and (ii) We show the inequality in the minimal framework required to state it: abelian metric groups.
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