  We obtain strong moment invariance principles for normalized multiple
iterated sums and integrals of the form
$\bbS_N^{(\nu)}(t)=N^{-\nu/2}\sum_{0\leq k_1<...<k_\nu\leq
Nt}\xi(k_1)\otimes\cdots\otimes\xi(k_\nu)$, $t\in[0,T]$ and
$\bbS_N^{(\nu)}(t)=N^{-\nu/2}\int_{0\leq s_1\leq...\leq s_\nu\leq
Nt}\xi(s_1)\otimes\cdots\otimes\xi(s_\nu)ds_1\cdots ds_\nu$, where
$\{\xi(k)\}_{-\infty<k<\infty}$ and $\{\xi(s)\}_{-\infty<s<\infty}$ are
centered stationary vector processes with some weak dependence properties. We
show, in particular, that (in both cases) the distribution of $\bbS^{(\nu)}_N$
is $O(N^{-\del})$-close, $\del>0$ in the Prokhorov and the Wasserstein metrics
to the distribution of certain stochastic processes $\bbW_N^{(\nu)}$
constructed recursively starting from $W_N=\bbW_N^{(1)}$ which is a Brownian
motion with covariances. This is done by constructing a coupling between
$\bbS_N^{(1)}$ and $\bbW_N^{(1)}$, estimating directly the moment variational
norm of $\bbS_N^{(\nu)}-\bbW_N^{(\nu)}$ for $\nu=1,2$ and extending these
estimates to $\nu>2$ relying on the rough paths theory.
  In the continuous time we work both under direct weak dependence assumptions
and also within the suspension setup which is more appropriate for applications
in dynamical systems.
