Convergence rate for stochastic consensus algorithms with time-varying noise statistics: Asymptotic normality

This paper studies consensus seeking over noisy networks with time-varying noise statistics. Stochastic approximation type algorithms can ensure consensus in mean square and with probability one. For performance evaluation, we examine the long term behavior of the approximation error which consists of two naturally defined components. We show that the two components and their sum are each asymptotically normal after being normalized by the square root of time. This, in turn, characterizes the convergence rate of the algorithm. We also give the asymptotic formula for the scaled error covariances.