  Given a weakly dependent stationary process, we describe the transition
between a Berry-Esseen bound and a second order Edgeworth expansion in terms of
the Berry-Esseen characteristic. This characteristic is sharp: We show that
Edgeworth expansions are valid if and only if the Berry-Esseen characteristic
is of a certain magnitude. If this is not the case, we still get an optimal
Berry-Esseen bound, thus describing the exact transition. We also obtain
(fractional) expansions given $3 < p \leq 4$ moments, where a similar
transition occurs. Corresponding results also hold for the Wasserstein metric
$W_1$, where a related, integrated characteristic turns out to be optimal. As
an application, we establish novel weak Edgeworth expansion and CLTs in $L^p$
and $W_1$. As another application, we show that a large class of high
dimensional linear statistics admit Edgeworth expansions without any smoothness
constraints, that is, no non-lattice condition or related is necessary. In all
results, the necessary weak-dependence assumptions are very mild. In
particular, we show that many prominent dynamical systems and models from time
series analysis are within our framework, giving rise to many new results in
these areas.
