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 = E S/n and q = 1 − p. Let 0 < t < q. In this paper, we extend the Hoeffding inequality [16, Theorem 1] , 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 < ∞. We also provide improvements comparable with the inequalities of Bentkus [5]. The independence of X _k can be replaced by supermartingale-type assumptions. Our methods can be extended to prove counterparts of other inequalities of Hoeffding [16] and Bentkus [5]. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No T-25/08.