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Partial updating of LMS filter coefficients is an effective method for reducing computational load and power consumption in adaptive filter implementations. This paper presents an analysis of convergence of the class of Sequential Partial Update LMS algorithms (S-LMS) under various assumptions and shows that divergence can be prevented by scheduling coefficient updates at random, which we call the Stochastic Partial Update LMS algorithm (SPU-LMS). Specifically, under the standard independence assumptions, for wide sense stationary signals, the S-LMS algorithm converges in the mean if the step-size parameter μ is in the convergent range of ordinary LMS. Relaxing the independence assumption, it is shown that S-LMS and LMS algorithms have the same sufficient conditions for exponential stability. However, there exist nonstationary signals for which the existing algorithms, S-LMS included, are unstable and do not converge for any value of μ. On the other hand, under broad conditions, the SPU-LMS algorithm remains stable for nonstationary signals. Expressions for convergence rate and steady-state mean-square error of SPU-LMS are derived. The theoretical results of this paper are validated and compared by simulation through numerical examples.