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Decoupling of Banach-valued multilinear forms in independent symmetric Banach-valued random variables
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  • Published: August 1987

Decoupling of Banach-valued multilinear forms in independent symmetric Banach-valued random variables

  • Terry R. McConnell1 &
  • Murad S. Taqqu2 

Probability Theory and Related Fields volume 75, pages 499–507 (1987)Cite this article

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  • 6 Citations

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Summary

Let E be a Banach space and Π: E→ℝ+ be symmetric, continuous and convex. Let {U i} and {r i} be independent sequences of random variables having, respectively, U(0, 1) and symmetric Bernoulli distributions, and let {U (j)i } and {r (j)i } for j=1, 2, ..., d be independent copies of these sequences. We prove two-sided inequalities between the quantities

$$E\Phi (\sum\limits_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i} \in \mathbb{Z}_ + ^d } {r_{i_1 } } ...r_{i_d } F_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i} } (U_{i_1 } ,...,U_{i_d } ))$$

and their “decoupled” versions

$$E\Phi (\sum\limits_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\iota } \in \mathbb{Z}_ + ^d } {r_{i_1 }^{(1)} ...r_{i_d }^{(d)} } F_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\iota } } (U_{i_1 }^{(1)} ,..., U_{i_d }^{(d)} ))$$

, for Bochner integrable F i : [0, 1]d→E. This generalizes results of Kwapień and of Zinn.

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References

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Author information

Authors and Affiliations

  1. Syracuse University, 200 Carnegie, 13244-1150, NY, Syracuse, USA

    Terry R. McConnell

  2. Boston University, 111 Cummington Street, 02215, Boston, MA, USA

    Murad S. Taqqu

Authors
  1. Terry R. McConnell
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  2. Murad S. Taqqu
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Additional information

Supported in part by NSF grant DMS 85-03775 and by the Sloan Foundation

Supported in part by NSF grant ECS 84-08524

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McConnell, T.R., Taqqu, M.S. Decoupling of Banach-valued multilinear forms in independent symmetric Banach-valued random variables. Probab. Th. Rel. Fields 75, 499–507 (1987). https://doi.org/10.1007/BF00320330

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  • Received: 21 January 1986

  • Issue Date: August 1987

  • DOI: https://doi.org/10.1007/BF00320330

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Keywords

  • Banach Space
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Bernoulli Distribution
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