  Given a $1$-cocycle $b$ with coefficients in an orthogonal representation, we
show that any finite dimensional summand of $b$ is cohomologically trivial if
and only if $\| b(X_n) \|^2/n$ tends to a constant in probability, where $X_n$
is the trajectory of the random walk $(G,\mu)$. As a corollary, we obtain
sufficient conditions for $G$ to satisfy Shalom's property $H_{\mathrm{FD}}$.
Another application is a convergence to a constant in probability of
$\mu^{*n}(e) -\mu^{*n}(g)$, $n\gg m$, normalized by its average with respect to
$\mu^{*m}$, for any finitely generated amenable group without infinite
virtually Abelian quotients. Finally, we show that the harmonic equivariant
mapping of $G$ to a Hilbert space obtained as an $U$-ultralimit of normalized
$\mu^{*n}- g \mu^{*n}$ can depend on the ultrafilter $U$ for some groups.
