Consensus Formation in a Two-Time-Scale Markovian System

This work analyzes distributed linear averaging within a connected network of sensors that each track the stationary distribution of an ergodic Markov chain with a slowly switching regime. Our approach is based on a two-time-scale stochastic approximation. A hyperparameter modeled as a Markov chain on a slower time-scale modulates the regime of each observed Markov chain. The average of all currently observed stationary distributions constitutes the average-consensus estimate to be reached by all sensors. Assuming the Markov chains do not share a common stationary distribution conditioned on their regime, then under the proposed linear averaging algorithm, the exchange graph conditions required for the sequence of sensor state values to converge weakly to the average-consensus are obtained. Estimation of a weighted average of all observed stationary distributions, not only the current ones, is proved feasible over a long-run time horizon, provided an additional communication condition holds. The sensor st...

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