In this article, we discuss some geometric infinitely divisible (gid) random variables using the Laplace exponents which are Bernstein functions and study their properties. The distributional properties and limiting behavior of the probability densities of these gid random variables at 0+ are studied. The autoregressive (AR) models with gid marginals are introduced. Further, the first order AR process is generalised to kth order AR process. We also provide the parameter estimation method based on conditional least square and method of moments for the introduced AR(1) processes.

翻译：本文利用拉普拉斯指数（即伯恩斯坦函数）讨论了几何无限可分随机变量，并研究其性质。研究了这些几何无限可分随机变量在0+处的分布特性及概率密度的极限行为。引入了具有几何无限可分边缘分布的自回归模型。进一步，将一阶自回归过程推广至k阶自回归过程。我们还基于条件最小二乘法与矩法，为所引入的AR(1)过程提供了参数估计方法。