Convergence properties of LMS adaptive estimators with unbounded dependent inputs

This note presents limit theorems for the behavior of adaptive estimators using the LMS algorithm when the driving or input sequence is a member of a broad class of random processes which are not necessarily almost surely bounded and are dependent over time. Convergence in distribution of the estimates is established in the stationary case while general nonstationary tracking is characterized in the nonstationary case. These results follow from the exponential convergence of the homogeneous algorithm which in turn follows from a strong limit theorem for infinite products of ergodic and mixing sequences of matrices.