Inferring the strength of conditional dependence and testing conditional independence are fundamental problems in statistics. A recent breakthrough by Azadkia and Chatterjee introduced, for the first time, a conditional dependence measure that equals $0$ if and only if the variables under study are conditionally independent, and equals $1$ if and only if they are conditionally perfectly dependent. They further proposed a computationally efficient and strongly consistent estimator, $T_n$, based on an ingenious use of ranks and nearest neighbors. Despite these attractive features, the asymptotic theory of $T_n$ has remained largely undeveloped. This paper closes that gap. We prove that, under general dependence, $T_n$ is asymptotically normal and its limiting variance admits a closed form. We also construct consistent variance estimators that are computationally efficient and implementable in $O(n\log n)$ time. Taken together with existing bias-correction methods, these results provide a complete inferential theory for $T_n$.

翻译：推断条件依赖的强度以及检验条件独立性是统计学中的基础问题。阿扎德基亚和查特吉近期的一项突破性工作首次引入了一种条件依赖度量，该度量在变量条件独立时等于0，在变量条件完全依赖时等于1。他们进一步基于对秩与最近邻的巧妙运用，提出了一种计算高效且强相合的估计量$T_n$。尽管这一估计量具有诸多吸引人的特性，但其渐近理论尚不完善。本文填补了这一空白。我们证明，在一般依赖条件下，$T_n$具有渐近正态性，且其极限方差存在闭合表达式。我们还构建了相合的方差估计量，这些估计量计算高效，可在$O(n\log n)$时间内实现。结合现有的偏差校正方法，这些结果为$T_n$提供了完整的推断理论。