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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A sharp inequality for the $p$-center of gravity of a random variable
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by David C. Cox PDF
Proc. Amer. Math. Soc. 93 (1985), 106-110 Request permission

Abstract:

Let $X$ be a real-valued random variable. For $p > 1$, define the $p$-center of gravity of $X$, ${C_p}(X)$, as the unique number $c$ which minimizes ${\left \| {X - c} \right \|_p}$. This paper exhibits a more-or-less explicit expression for the best constant $\gamma = {\gamma _p}$ in the inequality $\left | {E(X) - {C_p}(X)} \right | \leqslant \gamma {\left \| {X - {C_p}(X)} \right \|_p}$, and presents asymptotic formulas for ${\gamma _p}$ as $p \to 1,2$ and $+ \infty$, respectively. The definition of ${C_p}(X)$ is extended to variables taking values in an arbitrary Hilbert space $H$, and it is shown that ${\gamma _p}$ is not increased by this extension.
References
    D. C. Cox, Sharp inequalities for martingales, Doctoral Dissertation, Univ. of Rochester, 1979 (available from the author).
  • Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121
  • J. H. B. Kemperman, The general moment problem, a geometric approach, Ann. Math. Statist. 39 (1968), 93–122. MR 247645, DOI 10.1214/aoms/1177698508
  • F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
  • H. S. Witsenhausen, On performance bounds for uncertain systems, SIAM J. Control 8 (1970), 55–89. MR 0273738, DOI 10.1137/0308004
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 93 (1985), 106-110
  • MSC: Primary 60E15; Secondary 44A60
  • DOI: https://doi.org/10.1090/S0002-9939-1985-0766538-4
  • MathSciNet review: 766538