An extension of the Hoeffding inequality to unbounded random variables

AbstractLet S = X1 + ⋯ + Xn be a sum of independent random variables such that 0 ⩽ Xk ⩽ 1 for all k. Write p = E S/n and q = 1 − p. Let 0 < t < q. In this paper, we extend the Hoeffding inequality [16, Theorem 1] $$ \mathbb{P}\left\{ {S \geqslant nt + np} \right\} \leqslant H^n \left( {t,p} \right), {\rm H}\left( {t,p} \right) = \left( {\frac{p} {{p + t}}} \right)^{p + t} \left( {\frac{q} {{q - t}}} \right)^{q - t} , $$, to the case where Xk are unbounded positive random variables. Our inequalities reduce to the Hoeffding inequality if 0 ⩽ Xk ⩽ 1. Our conditions are Xk ⩾ 0 and E S < ∞. We also provide improvements comparable with the inequalities of Bentkus [5]. The independence of Xk can be replaced by supermartingale-type assumptions. Our methods can be extended to prove counterparts of other inequalities of Hoeffding [16] and Bentkus [5].