Weakly dependent chains with infinite memory

We prove the existence of a weakly dependent strictly stationary solution of the equation X t = F ( X t − 1 , X t − 2 , X t − 3 , … ; ξ t ) called a chain with infinite memory. Here the innovations ξ 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.