Some remarks on the ergodic theorem for $U$-statistics

TL;DR

This paper examines the convergence properties of second-order U-statistics with stationary ergodic data, identifying conditions for almost sure and L^1 convergence, and highlighting the necessity of centering and boundedness for convergence.

Contribution

It provides new sufficient conditions for the convergence of U-statistics under ergodic assumptions and illustrates cases where convergence fails without these conditions.

Findings

Sufficient conditions for almost sure convergence

Conditions for L^1 convergence of U-statistics

Counter-examples showing the need for centering and boundedness

Abstract

In this note, we investigate the convergence of a $U$-statistic of order two having stationary ergodic data. We will find sufficient conditions for the almost sure and $L^1$ convergence and present some counter-examples showing that the $U$-statistic itself might fail to converge: centering is needed as well as boundedness of $\sup_{j\geq 2}\mathbb{E}[|h(X_1,X_j)|]$.