  We obtain convergence rates (in the Levi-Prokhorove metric) in the functional
central limit theorem (CLT) for partial sums $S_n=\sum_{j=1}^{n}\xi_{j,n}$ of
triangular arrays $\{\xi_{1,n},\xi_{2,n},...,\xi_{n,n}\}$ satisfying some
mixing and moment conditions (which are not necessarily uniform in $n$). For
certain classes of additive functionals of triangular arrays of contracting
Markov chains (in the sense of Dobrushin) we obtain rates which are close to
the best rates obtained for independent random variables. In addition, we
obtain close to optimal rates in the usual CLT and a moderate deviations
principle and some Rosenthal type inequalities. We will also discuss
applications to some classes of local statistics (e.g. covariance estimators),
as well as expanding non-stationary dynamical systems, which can be reduced to
non-uniformly mixing triangular arrays by an approximation argument. The main
novelty here is that our results are obtained without any assumptions about the
growth rate of the variance of $S_n$. The result are obtained using a certain
type of block decomposition, which, in a sense, reduces the problem to the case
when the variance of $S_n$ is "not negligible" in comparison with the (new)
number of summands.
