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)$X_t=F(X_{t-1},X_{t-2},X_{t-3},\dots ;{\xi }_t)$ called a chain with infinite memory. Here the innovations  _ξt${\xi }_t$ constitute an independent and identically distributed sequence of random variables. The function F$F$ 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$F$ 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.