$U$-Statistic Processes: A Martingale Approach

Abstract

For i.i.d. data $X_1,\cdots, X_n$ and a kernel $h$, the associated $U$-statistic process is defined as $U_n (u, v) = \frac{1}{n(n-1)} \sum_{1\leq i\neq j\leq n} h(X_i, X_j)1_{\{X_i\leq u,X_j\leq \nu\}}.$ Variants of these processes occur, for example, in the representation of the product-limit estimator of a lifetime distribution for censored/truncated data or in trimmed $U$-statistics. We derive an almost sure representation of $U_n$ under weak moment assumptions on $h$. Proofs rely on a proper decomposition of the remainder term into strong two-parameter martingales.