Addendum to “An extension of an inequality of Hoeffding to unbounded random variables”: The non-i.i.d. case

Abstract

Let Sn = X1 + ⋯ + Xn be a sum of independent random variables such that 0 ⩽ Xk ⩽ 1 for all k. Write {ie237-01} and q = 1 − p. Let 0 < t < q. In our recent paper [3], we extended the inequality of Hoeffding ([6], Theorem 1) {fx237-01} to the case where Xk are unbounded positive random variables. It was assumed that the means {ie237-02} of individual summands are known. In this addendum, we prove that the inequality still holds if only an upper bound for the mean {ie237-03} is known and that the i.i.d. case where {ie237-04} dominates the general non-i.i.d. case. Furthermore, we provide upper bounds expressed in terms of certain compound Poisson distributions. Such bounds can be more convenient in applications. Our inequalities reduce to the related Hoeffding inequalities if 0 ⩽ Xk ⩽ 1. Our conditions are Xk ⩾ 0 and {ie237-05}. In particular, Xk can have fat tails. We provide as well improvements comparable with the inequalities in Bentkus [2]. The independence of Xk can be replaced by super-martingale type assumptions. Our methods can be extended to prove counterparts of other inequalities in Hoeffding [6] and Bentkus