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

Let S-n = X-1 + X-n be a sum of independent random variables such that 0 <= X-k <= 1 for all k. Write p = {ESn/n and q = 1 - p. Let 0 < t < q. In our recent paper [3], we extended the inequality of Hoeffding ([6], Theorem 1), Theorem) to the case where X-k are unbounded positive random variables. It was assumed that the means mu(k) = EX(k)of individual summands are known. In this addendum, we prove that the inequality still holds if only an upper bound for the mean E S-n is known and that the i.i.d. case where E X-k = ESn < infinity. In particular, X-k can have fat tails. 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 <= X-k <= 1. Our conditions are X-k >= 0 and E S-n < infinity. In particular, X-k can have fat tails. We provide as well improvements comparable with the inequalities in Bentkus [2]. The independence of X-k can be replaced by super-martingale type assumptions. Our methods can be extended to prove counterparts of other inequalities in Hoeffding [6] and Bentkus [1], [2].