  We establish a quantitative normal approximation result for sums of random
variables with multilevel local dependencies. As a corollary, we obtain a
quantitative normal approximation result for linear functionals of random
fields which may be approximated well by random fields with finite dependency
range. Such random fields occur for example in the homogenization of linear
elliptic equations with random coefficient fields. In particular, our result
allows for the derivation of a quantitative normal approximation result for the
approximation of effective coefficients in stochastic homogenization in the
setting of coefficient fields with finite range of dependence. The proof of our
normal approximation theorem is based on a suitable adaption of Stein's method
and requires a different estimation strategy for terms originating from
long-range dependencies as opposed to terms stemming from short-range
dependencies.
