Two-Sample U-Statistic Processes for Long-Range Dependent Data

TL;DR

This paper studies the asymptotic behavior of two-sample U-statistics in long-range dependent data, introducing two approaches that handle a wide class of kernels relevant for change point detection.

Contribution

It develops two analytical methods for the asymptotic analysis of U-statistics under long-range dependence, broadening the scope of kernels that can be analyzed.

Findings

Two approaches for asymptotic distribution analysis are proposed.

The methods cover all commonly used kernels in applications.

Results facilitate change point testing in dependent data.

Abstract

Motivated by some common-change point tests, we investigate the asymptotic distribution of the U-statistic process $U_n(t)=\sum_{i=1}^{[nt]}\sum_{j=[nt]+1}^n h(X_i,X_j)$, $0\leq t\leq 1$, when the underlying data are long-range dependent. We present two approaches, one based on an expansion of the kernel $h(x,y)$ into Hermite polynomials, the other based on an empirical process representation of the U-statistic. Together, the two approaches cover a wide range of kernels, including all kernels commonly used in applications.