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This paper studies the interaction of automata of size m. We characterise statistical properties satisfied by random plays generated by a correlated pair of automata with m states each. We show that in some respect the pair of automata can be identified to a more complex automaton of size comparable to m log m. We investigate implications of these results on the correlated min-max value of repeated games played by automata.doi:10.1007/s00182-015-0521-7 fatcat:4h52rjqbjvg35h44aootyht47a