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We prove that any strongly mixing action of a countable abelian group on aprobability space has higher order mixing properties. This is achieved viaintroducing and utilizing $\mathcal R$-limits, a notion of convergence which isbased on the classical Ramsey Theorem. $\mathcal R$-limits are intrinsicallyconnected with a new combinatorial notion of largeness which is similar to buthas stronger properties than the classical notions of uniform density one andIP$^*$. While the main goal of this paper is to establish a$\textit{universal}$ property of strongly mixing actions of countable abeliangroups, our results, when applied to $\mathbb Z$-actions, offer a new way ofdealing with strongly mixing transformations. In particular, we obtain severalnew characterizations of strong mixing for $\mathbb Z$-actions, including aresult which can be viewed as the analogue of the weak mixing of all ordersproperty established by Furstenberg in the course of his proof of Szemer\'edi'stheorem. We also demonstrate the versatility of $\mathcal R$-limits byobtaining new characterizations of higher order weak and mild mixing foractions of countable abelian groups.

Strongly mixing systems are almost strongly mixing of all orders