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In this work, we study stability of distributed filtering of Markov chainswith finite state space, partially observed in conditionally Gaussian noise. Weconsider a nonlinear filtering scheme over a Distributed Network of Agents(DNA), which relies on the distributed evaluation of the likelihood part of thecentralized nonlinear filter and is based on a particular specialization of theAlternating Direction Method of Multipliers (ADMM) for fast average consensus.Assuming the same number of consensus steps between any two consecutive noisymeasurements for each sensor in the network, we fully characterize a minimalnumber of such steps, such that the distributed filter remains uniformly stablewith a prescribed accuracy level, {\varepsilon} \in (0,1], within a finiteoperational horizon, T, and across all sensors. Stability is in the sense ofthe \ell_1-norm between the centralized and distributed versions of theposterior at each sensor, and at each time within T. Roughly speaking, our mainresult shows that uniform {\varepsilon}-stability of the distributed filteringprocess depends only loglinearly on T and (roughly) the size of the network,and only logarithmically on 1/{\varepsilon}. If this total loglinear bound isfulfilled, any additional consensus iterations will incur a fully quantifiedfurther exponential decay in the consensus error. Our bounds are universal, inthe sense that they are independent of the particular structure of the GaussianHidden Markov Model (HMM) under consideration.

Uniform {\varepsilon}-Stability of Distributed Nonlinear Filtering over DNAs: Gaussian-Finite HMMs