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We study the asymptotic behaviour of the cross-variation of two-dimensionalprocesses having the form of a Young integral with respect to a fractionalBrownian motion of index $H \textgreater{} 1/ 2$. When $H$ is smaller than orequal to $3 / 4$, we show asymptotic mixed normality. When $H$ is strictlybigger than $3/4$, we obtain a limit that is expressed in terms of thedifference of two independent Rosenblatt processes.

Cross-variation of Young integral with respect to long-memory fractional Brownian motions