  We consider the unique measure of maximal entropy of an automorphism of a
compact K{\"a}hler manifold with simple action on cohomology. We show that it
is exponentially mixing of all orders with respect to H{\"o}lder observables.
It follows that the Central Limit Theorem (CLT) holds for these observables. In
particular, our result applies to all automorphisms of compact K{\"a}hler
surfaces with positive entropy.
