In this paper we clarify the relationships between the asymptotic convergence rates of recursive identification schemes, and local tests introduced by Basseville and Benveniste. First, we show how to derive convergence rates for identification procedures, from the central limit theorem used for the design of local tests. Second, we take advantage of this nonstandard route in order to establish laws of large numbers, and central limit theorems using simple proofs and weak assumptions, for stationary systems.

An interesting byproduct of this study is the following, practical, one. In general, the practical estimation of convergence rates for recursive stochastic algorithms is a difficult task (with the exception of linear regression or AR/ARMAX modelling). The design of local tests is much easier. And it turns out that, once the local test statistics is available, one can use it to derive an empirical estimate of the above mentioned convergence rate. An interesting application of this idea is Instrumental Variable and subspace identification methods, for which related local tests are available.

Keywords : Identification, recursive identification, adpative algorithms, convergence rates, local tests, M-estimators.