The Khinchin-Kahane and Levy inequalities for abelian metric groups, and transfer from normed (abelian semi)groups to Banach spaces

Abstract

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.