On the type hypothesis for the strong law of large numbers (Q1087219)

It is proved that for any separable Banach space B, if \(\{X_ n\); \(n\in {\mathbb{N}}\}\) is a sequence of i.i.d. B-valued random vectors with \(EX_ 1=0\) and \(1<\alpha <2\) then the following conditions are equivalent: \[ (i)\quad E\| X_ 1\|^{\alpha}<\infty; \] \[ (ii)\quad \| S_ n\| \cdot n^{-1/\alpha}\to 0\quad a.s.\quad as\quad n\to \infty; \] \[ (iii)\quad E\| S_ n\|^{\alpha}=o(n),\quad n\to \infty; \] \[ (iv)\quad \sum^{\infty}_{n=1}n^{-1}P\{\| S_ n\| >\epsilon n^{1/\alpha}\}<\infty \text{ for every } \epsilon >0. \] This means that in such a context the generalizations of the Pyke-Root, Baum-Katz and Marcinkiewicz-Zygmund theorems are equivalent.