Least mean square algorithms with Markov regime-switching limit

We analyze the tracking performance of the least mean square (LMS) algorithm for adaptively estimating a time varying parameter that evolves according to a finite state Markov chain. We assume the Markov chain jumps infrequently between the finite states at the same rate of change as the LMS algorithm. We derive mean square estimation error bounds for the tracking error of the LMS algorithm using perturbed Lyapunov function methods. Then combining results in two-time-scale Markov chains with weak convergence methods for stochastic approximation, we derive the limit dynamics satisfied by continuous-time interpolation of the estimates. Unlike most previous analyzes of stochastic approximation algorithms, the limit we obtain is a system of ordinary differential equations with regime switching controlled by a continuous-time Markov chain. Next, to analyze the rate of convergence, we take a continuous-time interpolation of a scaled sequence of the error sequence and derive its diffusion limit. Somewhat remarkably, for correlated regression vectors we obtain a jump Markov diffusion. Finally, two novel examples of the analysis are given for state estimation of hidden Markov models (HMMs) and adaptive interference suppression in wireless code division multiple access (CDMA) networks.