  In this note we consider stochastic differential equations driven by
fractional Brownian motions (fBm) with Hurst parameter $H>1/3$. We prove that
the corresponding modified Euler scheme and its Malliavin derivatives are
integrable, uniformly with respect to the step size $n$. Then we use the
integrability results to derive the convergence rate in law
$n^{1-4H+\varepsilon} $ for the Euler scheme. The proof for integrability is
based on a nontrivial generalization (to quadratic functionals of the fBm) of a
now classical greedy sequence argument laid out by Cass, Litterer and Lyons.
The proof of weak convergence applies Malliavin calculus and some upper-bound
estimates for weighted random sums.

Probability