Some remarks on the ergodic theorem for $U$-statistics
Résumé
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 $\sup_{j\geq 2}\mathbb E\left[\lvert h(X_1,X_j) \rvert\right]$.
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