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On inequalities of the Tchebychev type

Published online by Cambridge University Press:  24 October 2008

J. F. C. Kingman
J. F. C. Kingman
Affiliation:
Pembroke College, Cambridge
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Extract

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 1 , January 1963 , pp. 135 - 146
Copyright
Copyright © Cambridge Philosophical Society 1963

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References

REFERENCES

(1)Craig, C. C.On the Tchebycheff inequality of Bernstein. Ann. Math. Statist. 4 (1933), 94–102.CrossRefGoogle Scholar
(2)Eggleston, H. G.Convexity (Cambridge, 1958).Google Scholar
(3)Hoeffding, W.The extrema of the expected value of a function of independent random variables. Ann. Math. Statist. 26 (1955), 268–275.CrossRefGoogle Scholar
(4)Isii, K.On a method for generalizations of Tchebycheff's inequality. Ann. Inst. Statist. Math. Tokyo, 10 (1959), 65–88.Google Scholar
(5)Kingman, J. F. C. On queues in heavy traffic (to appear).Google Scholar
(6)Mallows, C. L.Generalizations of Tchebycheff's inequalities. J. Roy. Statist. Soc. Ser. B, 18 (1956), 139–176.Google Scholar
(7)Marshall, A. W. and Olkin, I.Multivariate Chebyshev inequalities. Ann. Math. Statist. 31 (1960), 1001–1014.CrossRefGoogle Scholar
(8)Whittle, P.A multivariate generalization of Tchebichev's inequality. Quart. J. Math. Oxford Ser. (2), 9 (1958), 232–240.Google Scholar
(9)Whittle, P.Continuous generalizations of Tchebichev's inequality. Teor. Veroyatnost. i Primenen, 3 (1958), 386–395.Google Scholar