Some remarks on the ergodic theorem for U-statistics

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 finiteness of sup j≥2 𝔼[|h(X 1 ,X j )|].