  We study some properties of the randomized series and their applications to
the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is
shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only
if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for
various convexity properties in Banach spaces are also studied. It is proved
that a Banach lattice $E$ is uniformly monotone if and only if its
$p$-convexification $E^{(p)}$ is uniformly convex and that a K\"othe function
space $E$ is upper locally uniformly monotone if and only if its
$p$-convexification $E^{(p)}$ is midpoint locally uniformly convex.
