Limit theorems of Azadkia-Chatterjee's conditional graph correlation

Abstract

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 $T_n$.