Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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] Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing; Graduate Texts in Mathematics, No. 25. MR 0367121
  • [3] J. H. B. Kemperman, The general moment problem, a geometric approach, Ann. Math. Statist 39 (1968), 93–122. MR 0247645,
  • [4] F. W. J. Olver, Asymptotics and special functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR 0435697
  • [5] H. S. Witsenhausen, On performance bounds for uncertain systems, SIAM J. Control 8 (1970), 55–89. MR 0273738

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60E15, 44A60

Retrieve articles in all journals with MSC: 60E15, 44A60

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society