  The central limit theorem (CLT) is one of the most fundamental results in
probability; and establishing its rate of convergence has been a key question
since the 1940s. For independent random variables, a series of recent works
established optimal error bounds under the Wasserstein-p distance (with p>=1).
In this paper, we extend those results to locally dependent random variables,
which include m-dependent random fields and U-statistics. Under conditions on
the moments and the dependency neighborhoods, we derive optimal rates in the
CLT for the Wasserstein-p distance. Our proofs rely on approximating the
empirical average of dependent observations by the empirical average of i.i.d.
random variables. To do so, we expand the Stein equation to arbitrary orders by
adapting the Stein's dependency neighborhood method. Finally we illustrate the
applicability of our results by obtaining efficient tail bounds.
