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Weak $L^{1}$ norms of random sums

Author: Paul Alton Hagelstein
Journal: Proc. Amer. Math. Soc. 133 (2005), 2327-2334
MSC (2000): Primary 42B35
Published electronically: March 4, 2005
MathSciNet review: 2138875
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Abstract: Let $\left\{g_{j}\right\}$ denote a sequence of measurable functions on $\mathbf{R}^{n}$, and let $\left\Vert\cdot\right\Vert _{WL^{1}}$ denote the weak $L^{1}$ norm. It is shown that

\begin{displaymath}\left\Vert\mathbb{E}\left(\left\vert\sum_{j=1}^{N} \epsilon_{... ... \lesssim \sum_{j=1}^{N} \left\Vert g_{j}\right\Vert _{WL^{1}},\end{displaymath}

where $\left\{\epsilon_{j}\right\}$ is a sequence of independent random variables taking on values $+1$ and $-1$ with equal probability. Moreover, it is shown that

\begin{displaymath}\left\Vert \mathbb{E}\left(\left\vert\sum_{j=1}^{N} \epsilon_... ...rt\sum_{j=1}^{N} \epsilon_{j}g_{j}\right\Vert _{WL^{1}}\right).\end{displaymath}

The paper concludes by providing an example indicating that, if $\left\Vert g_{1}\right\Vert _{WL^{1}}$ $= \cdots = \left\Vert g_{N}\right\Vert _{WL^{1}} = 1$, then the estimate

\begin{displaymath}\mathbb{E}\left(\left\Vert\sum_{j=1}^{N}\epsilon_{j}g_{j}\right\Vert _{WL^{1}}\right) \lesssim N \log N\end{displaymath}

is the best possible.

References [Enhancements On Off] (What's this?)

  • 1. E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc. 140 (1969), 35-54. MR 0241685 (39:3024)
  • 2. A. Zygmund, Trigonometric Series, Cambridge University Press, 1959.MR 0107776 (21:6498)

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Additional Information

Paul Alton Hagelstein
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798

Received by editor(s): November 21, 2003
Published electronically: March 4, 2005
Additional Notes: The author’s research was partially supported by the Baylor University Summer Sabbatical Program.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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