独立性并非总是可一致检验的
Independence Is Not Always Consistently Testable
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中文总结 AI 辅助
该研究证明，从有限观测中无法一致检验联合平稳遍历二元过程的坐标过程间的独立性，因假阳性概率上极限至少为1/2，其证明结合了对角构造与稀有标记。
中文摘要 AI 辅助
我们研究从有限观测中检验联合平稳遍历二元过程的坐标过程间独立性的问题。我们证明，不存在逐点概率一致的检验：任何对所有相关律的功效趋于1的方法，在某些独立平稳遍历律上，针对无穷多样本量的假阳性概率均非零。定量而言，对某些坐标过程独立的联合平稳遍历律，假阳性概率的上极限至少为1/2。因此，即便在平稳性与遍历性下，也无法通过越来越长的有限样本一致判定独立性。该证明结合了对角构造与稀有标记，这些标记在选定尺度上产生可检测的相关性，同时收敛到坐标独立的极限过程。
英文摘要
We study the problem of testing independence between the coordinate processes of a jointly stationary ergodic binary process from finite observations. We prove that no test is pointwise consistent in probability: any procedure whose power tends to one against every dependent law must have nonvanishing false-positive probability on some independent stationary ergodic law along infinitely many sample sizes. Quantitatively, for some jointly stationary ergodic law with independent coordinate processes, the false-positive probability has limsup at least $1/2$. Thus, even under stationarity and ergodicity, independence cannot be consistently decided from increasingly long finite samples. The proof combines a diagonal construction with rare markers that create detectable dependence at selected scales while converging to a limiting process whose coordinates are independent.
