Few results are available in the literature on failure time (in contrast with long time average) analysis of the least mean square (LMS) adaptive algorithms (filters). Such analysis is extremely important when failure of the algorithm could cause failure of the entire system in which they are employed. A system may fail upon the first occurrence of a large error (any value far from 0), or upon staying far from 0 for a period of time (a clump of large errors). We use a Poisson approximation to study excursions (failure) of the LMS algorithm and its three signed variants. The distribution and the mean of the first excursion are approximated with the mean and distribution of an exponential distribution. The number of excursions in n units of time is approximated by a Poisson distribution. We also approximate the mean of an excursion length (clump size). The approximations are derived asymptotically as the excursion-defining set converges to the empty set, and as the algorithm step size converges to 0.

This paper analyzes excursions of adaptive algorithms (such as the LMS) with a lattice state space. Under certain conditions on the input and disturbance statistics, the parameter estimate error forms a Markov chain. The approximations are valid if this chain has a strong tendency toward an equilibrium point. The distribution of the number of excursions in n units of time is approximated by a Poisson distribution. The mean and distribution of the time of the occurrence of the first excursion are approximated by those of an exponential distribution. Expressions for the error in the approximations are also derived. The approximations are shown to hold asymptotically as the excursion-defining set converges to the empty set. All the parameters required for the approximations and all expressions for the error in the approximations are calculable in a relatively straightforward manner.