A One-Sided Analog of Kolmogorov's Inequality

Abstract

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.