Discretely sampled signals and the rough Hoff path

Sampling a $d$-dimensional continuous signal (say a semimartingale) $X:[0,T] \rightarrow \mathbb{R}^d$ at times $D=(t_i)$, we follow the recent papers [Gyurko-Lyons-Kontkowski-Field-2013] and [Lyons-Ni-Levin-2013] in constructing a lead-lag path; to be precise, a piecewise-linear, axis-directed process $X^D: [0,1] \rightarrow
\mathbb{R}^{2d}$ comprised of a past and future component. Lifting $X^D$ to its natural rough path enhancement, we can consider the question of convergence as
the latency of our sampling becomes finer.