  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},...;\xi_t)$ called {\em chain with
infinite memory}. Here the {\em 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 and the rate of decay
of the Lipschitz coefficients of the function $F$. With the help of the weak
dependence properties, we derive Strong Laws of Large Number, a Central Limit
Theorem and a Strong Invariance Principle.
