Limit Theory for U-Statistics under Clustered and Weakly Dependent Data

Abstract

This paper develops asymptotic theory and feasible inference for unbounded-kernel order-k U-statistics under clustered sampling and weakly dependent time-series sampling. The analysis first builds the complete order-2 pipeline, moving from clustered data to exact m-dependence and then to near-epoch dependence. The same logic is subsequently extended to general order k greater than or equal to 2. Under clustered sampling, the theory allows arbitrary within-cluster dependence and growing, unbalanced cluster sizes. Under weak dependence, an i.i.d.-based approximating sequence carries the exact-m theory to near-epoch-dependent processes. The common combinatorial device is a vertical rearrangement, which isolates sampling-generic tuples, where the first-order Hoeffding projection is analysed, from collision terms and higher-order degenerate remainders, which are controlled explicitly. Cluster-robust and HAC estimators of the covariance of the first-order projection, needed for feasible inference, are shown to be consistent.