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Stationarity and ergodicity conditions are given, and an easy-to-check general sufficient stationarity condition based on a tailor-made norm is introduced. Moreover, it is shown that causality of all individual regimes is neither a necessary nor a sufficient criterion for strict negativity of the associated Lyapunov exponent.<\/jats:p><jats:p>Finiteness of moments is also considered and geometric ergodicity and strong mixing are proven. 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