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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0766538-4
PII: S 0002-9939(1985)0766538-4
Article copyright: © Copyright 1985 American Mathematical Society