On Hoeffding's inequalities.
Let \(X_j, 1 \leq j \leq n\), be bounded independent random variables and define \(M_n=X_1+\dots+X_n.\) The author proves that \(P(M_n \geq x) \leq c P(S_n \geq x)\) where \(c\) is an absolute constant and \(S_n= \varepsilon_1+\dots+\varepsilon_n\) is a sum of independent identically distributed Bernoulli random variables. The inequality holds for those real numbers \(x\) at which the function \(B(x)=P(S_n\geq x)\) has a downward jump. For other values of \(x,\) the inequality holds provided that the function \(B(x)=P(S_n \geq x)\) is linearly or log-linearly interpolated between adjacent jump points. The results are extended to martingales with bounded differences.
bounded differences and random variables
scientific article