Stable Polynomials and Sums of Dependent Bernoulli Random Variables: Application to Hoeffding Inequalities

We give, in this paper, a characterization of the independent representation in law for a sum of dependent Bernoulli random variables. This characterization is related to the stability property of the probability-generating function of this sum, which is a polynomial with positive coefficients. As an application, we give a Hoeffding inequality for a sum of dependent Bernoulli random variables when its probability-generating function has all its roots with negative real parts. Some sufficient conditions on the law of the sum of dependent Bernoulli random variables guaranteeing the negativity of the real parts of the roots are discussed. This paper generalizes some results in Liggett (Stoch Process Appl 119:1–15, 2009).