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

2005/04/22 by Jan van Neerven, Mark Veraar, van Neerven, Jan +1
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Functional Equations Stability Results #math.FA #math.PR #msc:46B09 #msc:46C15 #msc:47B10

paper · pdf · doi:10.48550/arxiv.math/0504452

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

arxiv created 2005/04/22 · arxiv updated 2009/12/01

Abstract

Let X be a Banach space and let (ξ_j)_{j≥ 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 (\E‖∑_j=1ⁿ ξ_j f(x_j)‖²)^\frac12 ≤ K \n f\n_{\rm Lip} (\E‖∑_j=1ⁿ ξ_j x_j‖²)^\frac12 for all Lipschitz functions f:X→ X satisfying f(0)=0 and all finite sequences x₁,...,x_n in X. (2). X is isomorphic to a Hilbert space.

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