Probability distribution of steady-state errors and adaptation over networks

In this work, we derive a near-optimal combination rule for adaptation over networks. To do so, we first establish a useful result pertaining to the steady-state distribution of the estimator of an LMS filter. Specifically, under small step-sizes and some conditions on the data, we show that the steady-state estimator is approximately Gaussian and provide an expression for its covariance matrix. The result is subsequently used to show that the maximum ratio combining rule over networks, which is used to combine the estimators across neighbors within a network, is near optimal in the minimum variance unbiased sense. The result suggests a rule for combining the estimators within neighborhoods that can lead to improved mean-square error performance.