``Behind every limit theorem, there is an inequality'' said Kolmogorov. We say ``for every inequality, there is an approximate inequality under approximate regularity conditions.'' Suppose $X, X'$ are independent and identically distributed random variables. Then $X \le X'$ with a probability of at least $1/2$, irrespective of the underlying (common) distribution. One can ask what happens to the probability if $X, X'$ are independent but not identically distributed. It should be approximately $1/2$ if the distributions are approximately equal. Similarly, what if the random variables are dependent? It should, again, be approximately $1/2$ if the random variables are approximately independent. We explore an extension of this probability inequality involving order statistics and develop approximate versions of such an inequality under violations of independence and identical distribution assumptions. We further show that this inequality can be used as a basis to prove asymptotic validity of bootstrap/subsampling, finite-sample validity of conformal prediction, permutation tests, and asymptotic validity of rank tests without group invariance. Specifically, in the context of resampling inference, our results can be seen as a finite-sample instantiation of some results by Peter Hall and yield an alternative ``cheap bootstrap'' that applies to high-dimensional data.

翻译：柯尔莫哥洛夫曾说：“每个极限定理背后都对应一个不等式。”而我们则提出：“每个不等式在近似正则条件下都存在一个近似不等式。”设 $X, X'$ 为独立同分布随机变量，则无论其（公共）分布如何，$X \le X'$ 的概率至少为 $1/2$。一个自然的问题是：若 $X, X'$ 独立但非同分布，该概率将会如何变化？当分布近似相等时，该概率应近似为 $1/2$。类似地，若随机变量存在相依性，当它们近似独立时，该概率同样应近似为 $1/2$。本文探讨了该概率不等式在顺序统计量框架下的推广，并在违反独立性和同分布假设的条件下建立了该不等式的近似版本。我们进一步证明，该不等式可作为证明以下方法渐近有效性的基础：自助法/子抽样法、共形预测的有限样本有效性、置换检验，以及无需群不变性假设的秩检验。特别地，在重抽样推断情境下，我们的结果可视为彼得·霍尔某些结论的有限样本实现，并衍生出适用于高维数据的替代性“廉价自助法”。