Poisson approximation for excursions of adaptive algorithms with a lattice state space

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.