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The Khinchin-Kahane inequality is a fundamental result in the probabilityliterature, with the most general version to date holding in Banach spaces.Motivated by modern settings and applications, we generalize this inequality toarbitrary 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 tothe same one as in Banach spaces. This occurs via a "transfer principle" thathelps carry over questions involving normed metric groups and abelian normedsemigroups into the Banach space framework. This principle also extends thenotion of the expectation to random variables with values in arbitrary abeliannormed metric semigroups $\mathscr{G}$. We provide additional applications,including studying weakly $\ell_p$ $\mathscr{G}$-valued sequences and relatedRademacher series.  On a related note, we also formulate a "general" Levy inequality, with twofeatures: (i) It subsumes several known variants in the Banach spaceliterature; and (ii) We show the inequality in the minimal framework requiredto state it: abelian metric groups.

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