  In this article, an uniform discretization of stochastic integrals
$\int_{0}^{1} f'_-(B_t)\ud B_t$, with respect to fractional Brownian motion
with Hurst parameter $H \in (1/2,1)$, for a large class of convex functions $f$
is considered. In Statistics & Decisions, 27, 129-143, for any convex function
$f$, the almost sure convergence of uniform discretization to such stochastic
integral is proved. Here we prove $L^r$- convergence of uniform discretization
to stochastic integral. In addition, we obtain a rate of convergence. It turns
out that the rate of convergence can be brought as closely as possible to $H -
1/2$.
