In this paper, we investigate the cumulative distribution functions (CDFs) of the maximum and minimum of multivariate Poisson distributions with three dependence structures, namely, the common shock, comonotonic shock and thinning-dependence models. In particular, we formulate the definition of a thinning-dependent multivariate Poisson distribution based on Wang and Yuen (2005). We derive explicit CDFs of the maximum and minimum of the multivariate Poisson random vectors and conduct asymptotic analyses on them. Our results reveal the substantial difference between the three dependence structures for multivariate Poisson distribution and may suggest an alternative method for studying the dependence for other multivariate distributions. We further provide numerical examples demonstrating obtained results.

翻译：本文研究了具有三种依赖结构（即共同冲击、共单调冲击与稀释依赖模型）的多元泊松分布中最大值与最小值的累积分布函数。特别地，我们基于Wang和Yuen（2005）的研究，构建了稀释依赖型多元泊松分布的定义。我们推导了多元泊松随机向量最大值与最小值的显式累积分布函数，并对其进行了渐近分析。研究结果揭示了多元泊松分布三种依赖结构间的显著差异，这为研究其他多元分布的依赖性提供了替代方法。我们进一步通过数值算例验证了所得结论。