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 an d disturbance statistics, the parameter estimate error forms a Markov chain, The approximations are valid if this chain has a strong tendenc y toward an equilibrium point, The distribution of the number of excur sions in n units of time is approximated by a Poisson distribution, Th e mean and distribution of the time of the occurrence of the first exc ursion are approximated by those of an exponential distribution. Expre ssions for the error in the approximations are also derived, The appro ximations are shown to hold asymptotically as the excursion-defining s et converges to the empty set, All the parameters required for the app roximations and all expressions for the error in the approximations ar e calculable in a relatively straightforward manner.