We give qualitative and quantitative improvements to theorems which enable significance testing in Markov Chains, with a particular eye toward the goal of enabling strong, interpretable, and statistically rigorous claims of political gerrymandering. Our results can be used to demonstrate at a desired significance level that a given Markov Chain state (e.g., a districting) is extremely unusual (rather than just atypical) with respect to the fragility of its characteristics in the chain. We also provide theorems specialized to leverage quantitative improvements when there is a product structure in the underlying probability space, as can occur due to geographical constraints on districtings.

中文翻译：

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在马尔可夫链测试中将影响与重要性区分开

我们对定理进行了定性和定量的改进，从而可以在马尔可夫链中进行显着性检验，尤其着眼于实现强有力的，可解释的和统计上严格的政治主张主张的目标。我们的结果可用于在期望的显着性水平上证明，就其特征在链中的脆弱性而言，给定的马尔可夫链状态（例如，分区）是极其不寻常的（而不是非典型的）。当基础概率空间中存在产品结构时，我们也会提供专门用于利用定量改进的定理，这可能是由于分区的地理限制而发生的。