In this paper we obtain the limit distribution for partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixing between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between {\alpha}-stable distributions and NEF laws is established. We propose estimation of the parameters of the NEF models through the method of moments and also by the maximum likelihood method, which is performed via an Expectation-Maximization algorithm. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators and an empirical illustration on financial market is presented.

中文翻译：

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混合泊松随机和的收敛和推理

在本文中，我们获得了服从一类混合泊松分布的具有随机项数的部分和的极限分布。由此产生的弱极限是正态分布和指数族之间的混合，我们称之为正态指数族 (NEF) 定律。引入了一个新的稳定性概念，并建立了 {\alpha} 稳定分布和 NEF 定律之间的关系。我们建议通过矩方法和最大似然方法来估计 NEF 模型的参数，该方法通过期望最大化算法执行。蒙特卡罗模拟研究旨在检查建议的估计器的性能，并提供了金融市场的实证说明。