Convergence of the delayed normalized LMS algorithm with decreasing step size

In several practical applications of the LMS algorithm, including certain VLSI implementations, the coefficient adaptation can be performed only after some fixed delay. The resulting algorithm is known as the delayed LMS (DLMS) algorithm in the literature. Previous published analyses of this algorithm are based on mean and moment convergence under the independence assumption between successive input vectors. These analyses are interesting and give valuable insights into the convergence properties but, from a practical viewpoint, they do not guarantee the correct performance of the particular realization with which the user must live. We consider a normalized version of this algorithm with a decreasing step size μ(n) and prove the almost sure convergence of the nonhomogeneous algorithm, assuming a mixing input condition and the satisfaction of a certain law of large numbers