  In this paper we characterize the Lebesgue Bochner spaces
$L^p(\mathbb{R}^n,B)$, $1<p<\infty$, by using Littlewood-Paley $g$-functions in
the Hermite setting, provided that $B$ is a UMD Banach space. We use
$\gamma$-radonifying operators $\gamma (H,B)$ where
$H=L^2((0,\infty),\frac{dt}{t})$. We also characterize the UMD Banach spaces in
terms of $L^p(\mathbb{R}^n,B)$-$L^p(\mathbb{R}^n,\gamma (H,B))$ boundedness of
Hermite Littlewood-Paley $g$-functions.
