Let $\{X, X_n; n \geq 1\}$ be a sequence of i.i.d. random variables with values in a separable Banach spacc $B$ and set, for each $n, S_n = X_1 + \cdots + X_n$. We give necessary and sufficient conditions in order that $\lim\sup_{n\rightarrow\infty} n^{-1-(p/2)}(2L_2n)^{-(p/2)}\sum_{i=1}^n\|S_i\|^p < \infty \mathrm{a.s.},$ $\lim\sup_{n\rightarrow\infty} n^{-1-(p/2)}(2L_2n)^{-(p/2)}\sum_{i=0}^n\|S_n - S_i\|^p < \infty \mathrm{a.s.},$ where $p \geq 1$. Furthermore, the exact values of the above $\lim \sup$ are obtained. Some results are the extensions of Strassen's work to the vector settings and some are new even on the real line. The proofs depend on the construction of an independent sequence with values in $l_p(B)$ and appear as an illustration of the power of the limit law in Banach space.