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,...;[xi]t) called a chain with infinite memory. Here the innovations [xi]t constitute an independent and identically distributed sequence of random variables. The function F 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 F 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.