The laws of large numbers for identically distributed Banach space valued random variables

Let $0<\alpha<2$ and let $B_{\alpha}$ be an arbitrary Banach space if $0<\alpha\le 1$ and $B_{\alpha}$ be an $\alpha$-type space if $1<\alpha<2$ (definition of $\alpha$-type space see [1]); let $B_{\alpha}$ be separable when $\alpha\ge 1$. Without loss of generality we suppose that $\mathbf EX=0$ if $\mathbf E\|X\|<\infty$ where $X$ is Banach space valued random variable.
Theorem.{\it Let $0<\alpha<2$ and $\{X_n\}$ be a sequence of independent identically distributed $B_{\alpha}$-valued random variables, $S_n=X_1+\dots+X_n$. The following conditions are equivalent.}
I. $\mathbf E\|X_1\|^\alpha<\infty$.
II. $\|n^{-1/\alpha}S_n\|\to 0$ a. s., $n\to\infty$.
III. $\mathbf E\|S_n\|^{\alpha}=o(n)$, $n\to\infty$.
IV. $\displaystyle\sum_{n=1}^{\infty} n^{-1}\mathbf P\{\|S_n\|>\varepsilon n^{1/\alpha}\}<\infty$ for every $\varepsilon>0$.