Limit theorems of Azadkia-Chatterjee's conditional graph correlation

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, Tn, based on an ingenious use of ranks and nearest neighbors. Despite these attractive features, the asymptotic theory of Tn has remained largely undeveloped. This paper closes that gap. We prove that, under general dependence, Tn 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(nlog n) time. Taken together with existing bias-correction methods, these results provide a complete inferential theory for Tn.