The autoregressive model in a Banach space (ARB) allows to represent many continuous time processes used in practice. We consider the estimate of the operator of autocorrelation of ARB(1). The traditional estimate methods (maximum likelihood and least square) prove to be inadequate when parametric space is of infinite size. Grenander (1983} proposed to estimate the parameter on under space of size m in general finished, then study the consistency of this estimator when dimension m tends towards the infinite one with the numbers of observations at suitable speed. Let us note that more generallyit would be possible to use the f-divergences method. We define the least squares method like optimization problem in a Banach space when the operator is p-summable, p>1. We show consistency of the sieve estimator and we derive a central limit theorem for a strictly p-integral operator. We use the f-divergence dual representation to define the minimum f-divergences estimator. We limit our study here to theminimum of KL-divergence estimator (Kullback-Leibler divergence ). This estimator is that of the maximum likelihood. We show that it almost surely converges towards the true value of the parameter. The proof is based on the techniques of Geman and Hwang (1982), used for independent and identically distributed observations,that we adapt to the autoregression case.
