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).

在本文中，我们对依存的伯努利随机变量的总和进行了法律上独立代表的刻画。此特征与该和的概率生成函数的稳定性有关，该函数是具有正系数的多项式。作为一种应用，当其概率生成函数的所有根都为负实部时，我们对相关伯努利随机变量的和给出霍夫丁不等式。讨论了相关伯努利随机变量之和定律的一些充分条件，这些定律保证了根的实部为负。本文对Liggett中的一些结果进行了概括（Stoch Process Appl 119：1-15，2009年）。