Hypercontraction Methods in Moment Inequalities for Series of Independent Random Variables in Normed Spaces

We prove that if $(\theta_k)$ is a sequence of i.i.d. real random variables then, for $1 < q < p$, the linear combinations of $(\theta_k)$ have comparable $p$th and $q$th moments if and only if the joint distribution of $(\theta_k)$ is $(p, q)$-hypercontractive. We elaborate hypercontraction methods in a new proof of the inequality $\bigg(E\bigg\|\sum_i X_i\bigg\|^p\bigg)^{1/p} \leq C_p\bigg(E\big\|\sum_i X_i\bigg\| + \big(E\sup_i\|X_i\|^p\big)^{1/p}\bigg),$ where $(X_i)$ is a sequence of independent zero-mean random variables with values in a normed space, and $C_p \approx p/\ln p$.

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