  Let $X$ be a Banach space and let $(\xi_j)_{j\ge 1}$ be an i.i.d. sequence of
symmetric random variables with finite moments of all orders. We prove that the
following assertions are equivalent:
  (1). There exists a constant $K$ such that $$ \Bigl(\E\Big\|\sum_{j=1}^n
\xi_j f(x_j)\Big\|^2\Bigr)^{\frac12} \leq K \n f\n_{\rm Lip}
\Bigl(\E\Big\|\sum_{j=1}^n \xi_j x_j\Big\|^2\Bigr)^{\frac12} $$ for all
Lipschitz functions $f:X\to X$ satisfying $f(0)=0$ and all finite sequences
$x_1,...,x_n$ in $X$.
  (2). $X$ is isomorphic to a Hilbert space.
