ALMOST SURE ASYMPTOTIC PROPERTIES OF THE LEAST-SQUARES ESTIMATOR OF A VECTOR AUTOREGRESSIVE MODEL

For a vector valued autoregressive model which is control-lable, we prove the strong consistency of the least squares estimator except in the following "singular" case: there exists an eigensubspace of dimension greater-than-or-equal-to 2 associated to an eigenvalue of modulus > 1. In the singular case the consistency may fail. In the regular case, we precise the almost sure rate of convergence; we also study the predictor of this model and the empirical estimator of the covariance.