On the action of Lipschitz functions on vector-valued random sums

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 $$(\text{E}\parallel \sum _{j=1}^n{\xi }_{j}f(x_j){\parallel }^2{)}^{\frac{1}{2}}\le K\text{n}f{\text{n}}_{\mathrm{L}\mathrm{i}\mathrm{p}}(\text{E}\parallel \sum _{j=1}^n{\xi }_j{x}_j{\parallel }^2{)}^{\frac{1}{2}}$$ for all Lipschitz functions $f:X\to X$ satisfying $f(0)=0$ and all finite sequences $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n$ in $X$. (2). $X$ is isomorphic to a Hilbert space.