An extension of the Hoeffding inequality to unbounded random variables (Q946139)

Let \(S= X_1+ \dots+X_n\) be a sum of \(n\) independent random variables with \(E(S) < \infty.\) Let \(p= E(S)/n\) and \(q=1-p.\) Suppose \(0<t<q\). \textit{W. Hoeffding} [J. Am. Stat. Assoc. 58, 13--30 (1963; Zbl 0127.10602)] obtained an inequality on the tail probability \(P(S \geq nt+np)\) when the random variables \(X_k\), \(1 \leq k \leq n\), are such that \(0 \leq X_k \leq 1\), \(1 \leq k \leq n\). The author extends Hoeffding's inequality for nonnegative random variables \(X_k\), \(1 \leq k \leq n\), with \(E(S) < \infty\). The results continue to hold when the independence of the random variables \(X_k\), \(1 \leq k \leq n\), is replaced by a supermartingale-type assumption.