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

We study the asymptotic behaviour of the cross-variation of two-dimensional processes having the form of a Young integral with respect to a fractional Brownian motion of index $H \textgreater{} 1/ 2$. When $H$ is smaller than or equal to $3 / 4$, we show asymptotic mixed normality. When $H$ is strictly bigger than $3/4$, we obtain a limit that is expressed in terms of the difference of two independent Rosenblatt processes.