Convergence of an adaptive linear estimation algorithm

In this work we prove the almost sure convergence of an adaptive linear estimator governed by a stochastic gradient algorithm with decreasing step size in the presence of correlated observations. Two complementary contributions are added to the famous 1977 Ljung theorem. First we drop the condition of nondivergence of the algorithm assumed by Ljung. While that condition can be ensured by adding a barrier, the convergence of the suitably bounded algorithm itself is not established even on the basis of Ljung theorem. Here, the barrier problem is overcome by proving that it is not necessary for the convergence. Our second contribution is to generalize the model describing the correlated observations. No state space model is used and no linear relationship between the observations and the signal to be estimated needs to be assumed. Instead we use a decreasing covariance model that agrees with a very wide class of practical applications.