Strong consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator (QMLE) are given for a general class of multidimensional causal processes. For particular cases already studied in the literature (for instance univariate or multivariate GARCH, ARCH, ARMA-GARCH processes) the assumptions required for establishing these results are often weaker than existing conditions. The QMLE asymptotic behavior is also given for numerous new examples of univariate or multivariate processes (for instance TARCH or NLARCH processes).corresponds to the BEKK representation of multivariate GARCH(q, q ′ ) defined by Engle and Kroner [13], see also Bollerslev [4]. Their natural generalization,defines the multivariate ARCH(∞) processes. If M θ ≡ I d , a process X satisfying relation (1.1) is a multivariate Non Linear AR(∞) process.Various methods can be employed to estimate the unknown parameter θ 0 . Maximum Likelihood Estimation (MLE) is a common one. Several authors studied the asymptotic behavior of MLE for particular cases of multivariate processes satisfying (1.1), see for instance Bollerslev and Wooldridge [5], Jeantheau [19] for multivariate GARCH(q, q ′ ) processes and Dunsmuir and Hannan [11], Mauricio [22] for multivariate ARMA processes. A proof of the efficiency of those estimators was obtained in Berkes and Horváth [1], in the case of one-dimensional GARCH(q, q ′ ). Even if the convergence rate

We prove the existence of a weakly dependent strictly stationary solution of the equation Xt = F (Xt−1, Xt−2, Xt−3, . . . ; ξt) called chain with infinite memory. Here the innovations ξt 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.2000 Mathematics Subject Classification. Primary 62M10; Secondary 91B62, 60K35, 60K99, 60F05, 60F99.