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June, 1960 A One-Sided Analog of Kolmogorov's Inequality
Albert W. Marshall
Ann. Math. Statist. 31(2): 483-487 (June, 1960). DOI: 10.1214/aoms/1177705912


It is well known (see e.g. [4] p. 198) that for every positive $\epsilon$ and every square integrable random variable $X$ with zero expectation, $P{X \geqq \epsilon} \leqq E (X^2)/\lbrack\epsilon^2 + E(X^2)\rbrack$. In this paper an inequality is obtained that generalizes this in the same way that Kolmogorov's inequality generalizes Chebyshev's inequality. The inequality is proved in Section 2 and an example is given to show that equality can be achieved. In Section 3 an extension to continuous parameter martingales is obtained, and a condition under which equality can be achieved is given.


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Albert W. Marshall. "A One-Sided Analog of Kolmogorov's Inequality." Ann. Math. Statist. 31 (2) 483 - 487, June, 1960.


Published: June, 1960
First available in Project Euclid: 27 April 2007

zbMATH: 0099.13105
MathSciNet: MR119229
Digital Object Identifier: 10.1214/aoms/1177705912

Rights: Copyright © 1960 Institute of Mathematical Statistics

Vol.31 • No. 2 • June, 1960
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