Weakly dependent chains with infinite memory

We prove the existence of a weakly dependent strictly stationary solution of the equation Xt=F(Xt−1,Xt−2,Xt−3,…;ξt)Xt=F(Xt−1,Xt−2,Xt−3,…;ξt) called a chain with infinite memory. Here the innovations  ξtξt constitute an independent and identically distributed sequence of random variables. The function FF takes values in some Banach space and satisfies a Lipschitz-type condition. We also study the interplay between the existence of moments, the rate of decay of the Lipschitz coefficients of the function FF and the weak dependence properties. From these weak dependence properties, we derive strong laws of large number, a central limit theorem and a strong invariance principle.