Almost sure convergence of the normed LMS algorithm with error feedback delay

In some applications of LMS type adaptive algorithms, it necessary to implement a variant of the algorithm with feedback delay in the weight update calculation. We consider the normed version of such an algorithm and show that the algorithm converges exponentially if the update gain parameter, μ, is sufficiently small. The result is first proved for inputs which satisfy a standard deterministic mixing condition, and then the development is extended to the case when the input may not be strictly mixing but is instead a stationary ergodic vector sequence with positive definite autocorrelation