  We derive Edgeworth expansions that describe corrections to the Gaussian
limiting behaviour of slow-fast systems. The Edgeworth expansion is achieved
using a semi-group formalism for the transfer operator, where a Duhamel-Dyson
series is used to asymptotically determine the corrections at any desired order
of the time scale parameter $\varepsilon$. The corrections involve integrals
over higher-order auto-correlation functions. We develop a diagrammatic
representation of the series to control the combinatorial wealth of the
asymptotic expansion in $\varepsilon$ and provide explicit expressions for the
first two orders. At a formal level, the expressions derived are valid in the
case when the fast dynamics is stochastic as well as when the fast dynamics is
entirely deterministic. We corroborate our analytical results with numerical
simulations and show that our method provides an improvement on the classical
homogenization limit which is restricted to the limit of infinite time scale
separation.
