  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 $\sup_{j\geq 2}\mathbb{E}[|h(X_1,X_j)|]$.
