Isoperimetry and integrability of the sum of independent Banach-space valued random variables (Q583702)

Considered is a sequence of independent mean-zero Banach space valued random variables \(\{X_ i\}\). The author obtained some new inequalities for the tails of the distributions of \(\| \sum_{i\leq n}X_ i\|\). In particular, for any \(p\geq 1\) the following moment inequality holds: \[ \| \sum_{i\leq n}X_ i\|_ p\leq (K_ p/(1+\log p))(\| \sum_{i\leq n}X_ i\|_ 1+\| \max_{i\leq n}\| X_ i\| \|_ p) \] where \(\| \cdot \|_ p\) is the standard \(L_ p\)- norm. An inequality of this kind was previously known in cotype 2 spaces only.