Near-epoch-dependent multivariate U-statistic extension

Establish a near-epoch-dependent multivariate extension of the general-order U-statistic central limit theory for strongly mixing sequences with bounded kernels satisfying an extended variation condition.

Background

The introduction reviews existing weak-dependence results for U-statistics. It notes that Fischer et al. established general-order U-statistic central limit theorems for strongly mixing sequences under bounded-kernel and extended-variation assumptions, but did not resolve the corresponding near-epoch-dependent multivariate case.

The unresolved direction concerns extending the general-order, multivariate limit theory from strongly mixing sequences to near-epoch-dependent processes. This is distinct from the paper’s own framework, which develops scalar-kernel results under an L2-NED formulation and an i.i.d.-based approximating sequence.

References

\citet{fischer2016multivariate} proves general-order $U$-statistic central limit theorems for strongly mixing sequences with bounded kernels satisfying an extended variation condition, and leaves the near-epoch-dependent multivariate extension as a conjectural direction.

— Limit Theory for U-Statistics under Clustered and Weakly Dependent Data  (2608.18443 - Tsyawo, 19 Aug 2026) in Section 1, paragraph reviewing related weak-dependence results

Extending the present theory to dependent blocks remains open.

— Strong Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in $r$-Largest Generalized Extreme-Value Models  (2610.21161 - Park et al., 15 Oct 2026) in Section 4.2, “Scope and extensions”