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
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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