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

We prove that if (θₖ) is a sequence of i.i.d. real random variables then, for $1 < q < p$, the linear combinations of (θₖ) have comparable pth and qth moments if and only if the joint distribution of (θₖ) is $(p, q)$-hypercontractive. We elaborate hypercontraction methods in a new proof of the inequality (E\|∑ᵢ Xᵢ\|ᵖ)1/p ≤ Cₚ(E\|∑ᵢ Xᵢ\| + (Eᵢ\|Xᵢ\|ᵖ)1/p), where (Xᵢ) is a sequence of independent zero-mean random variables with values in a normed space, and Cₚ ≈ p/ln p.