Weak convergence results are obtained for a sequential regression algorithm that arises in the identification of nonlinear, memoryless systems and the adaptive design of moving average filters. The algorithm is shown to be weakly consistent if the system input is a wide-sense stationary sequence of order four that satisfies certain covariance and fourth-cumulant conditions. The conditions are essentially asymptotic independence requirements that permit one to relax the (usually required) strict independence requirements on the input data.