A stronger version of the Chebyshev inequality. Let X1, X2,…, Xn be independent random variables, such that, for j=1, 2,…, n, Xj has expectation 0 and |Xj|≤M. Bernstein showed, in a 1926 paper, that, for all positive ε,, where Var(Xj) = σj2. See also Chebyshev inequality; Hölder inequality; Kolmogorov inequality; Markov inequality; Minkowski inequality.
Subjects: Probability and Statistics.