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

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 ${\sup }_{j\ge 2}\mathbb{E}[\mathrm{\mid }h(X_1,X_j)\mathrm{\mid }]$.