Khinchin inequality and Banach–Saks type properties in rearrangement-invariant spaces

Volume 191 / 2009

F. A. Sukochev, D. Zanin Studia Mathematica 191 (2009), 101-122 MSC: 46E30, 46B20. DOI: 10.4064/sm191-2-1


We study the class of all rearrangement-invariant ($=\,$r.i.) function spaces $E$ on $[0,1]$ such that there exists $0< q< 1$ for which $ \Vert \sum_{k=1}^n\xi_k\Vert _{E}\leq Cn^{q}$, where $\{\xi_k\}_{k\ge 1}\subset E$ is an arbitrary sequence of independent identically distributed symmetric random variables on $[0,1]$ and $C>0$ does not depend on $n$. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces $\exp(L_p)$, $p\ge 1$. We further apply our results to the study of Banach–Saks index sets in r.i. spaces.


  • F. A. SukochevSchool of Mathematics and Statistics
    University of New South Wales
    Kensington, NSW 2052, Australia
  • D. ZaninSchool of Computer Science
    Engineering and Mathematics
    Flinders University
    Bedford Park, SA 5042, Australia

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