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

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

2005-04-22

arxiv.org/abs/math/0504452v1

Mathematics

Functional Analysis

8 pages, to appear in Archiv der Mathematik (Basel)

Scientific paper

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.

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