An extension of the Hoeffding inequality to unbounded random variables

Let S = X-1 + ... + X-n be a sum of independent random variables such that 0 <= X-k <= 1 for all k. Write p = ES/n and q = 1 - p. Let 0 < t < q. In this paper, we extend the Hoeffding inequality [16, Theorem 1] P{S >= nt + np} <= H-n(t, p), H(t, p) = (p/p+t)(p+t) (q/q-t)(q-t), to the case where X-k are unbounded positive random variables. Our inequalities reduce to the Hoeffding inequality if 0 <= X-k <= 1. Our conditions are X-k >= 0 and E S < infinity. 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].