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A sharp inequality for the $ p$-center of gravity of a random variable

Author: David C. Cox
Journal: Proc. Amer. Math. Soc. 93 (1985), 106-110
MSC: Primary 60E15; Secondary 44A60
MathSciNet review: 766538
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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\Vert {X - c} \right\Vert _p}$. This paper exhibits a more-or-less explicit expression for the best constant $ \gamma = {\gamma _p}$ in the inequality $ \left\vert {E(X) - {C_p}(X)} \right\vert \leqslant \gamma {\left\Vert {X - {C_p}(X)} \right\Vert _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 [Enhancements On Off] (What's this?)

  • [1] D. C. Cox, Sharp inequalities for martingales, Doctoral Dissertation, Univ. of Rochester, 1979 (available from the author).
  • [2] E. Hewitt and K. Stromberg, Real and abstract analysis, Grad. Texts in Math., Vol. 25, Springer-Verlag, New York, 1975. MR 0367121 (51:3363)
  • [3] J. H. B. Kemperman, The general moment problem, a geometric approach, Ann. Statist. 39 (1968), 93-122. MR 0247645 (40:909)
  • [4] F. W. J. Olver, Asymptotics and special functions, Academic Press, New York, 1974. MR 0435697 (55:8655)
  • [5] H. S. Witsenhausen, On performance bounds for uncertain systems, SIAM J. Control Optim. 8 (1970), 55-89. MR 0273738 (42:8615)

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Article copyright: © Copyright 1985 American Mathematical Society

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