Asymptotic properties of statistical estimators play a significant role both in practice and in theory. However, many asymptotic results in statistics rely heavily on the independent and identically distributed (iid) assumption, which is not realistic when we have fixed designs. In this article, we build a roadmap of general procedures for deriving asymptotic properties under fixed designs and the observations need not to be iid. We further provide their applications in many statistical applications. Finally, we apply our results to Poisson regression using a COVID-19 dataset as an illustration to demonstrate the power of these results in practice.

翻译：统计估计量的渐进性质在实践和理论中均具有重要作用。然而，统计学中的许多渐进结果严重依赖于独立同分布假设，这在固定设计条件下并不符合实际。本文构建了一套在固定设计下推导渐进性质的通用流程框架，且观测值无需满足独立同分布条件。我们进一步提供了这些方法在多种统计应用中的实施方案。最后，我们通过COVID-19数据集上的泊松回归分析实例，展示了这些结论在实际应用中的有效性。