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This paper studies the convergence of the estimation error process and thecharacterization of the corresponding invariant measure in distributed Kalmanfiltering for potentially unstable and large linear dynamic systems. A gossipnetwork protocol termed Modified Gossip Interactive Kalman Filtering (M-GIKF)is proposed, where sensors exchange their filtered states (estimates and errorcovariances) and propagate their observations via inter-sensor communicationsof rate $\overline{\gamma}$; $\overline{\gamma}$ is defined as the averagednumber of inter-sensor message passages per signal evolution epoch. Thefiltered states are interpreted as stochastic particles swapped through localinteraction. The paper shows that the conditional estimation error covariancesequence at each sensor under M-GIKF evolves as a random Riccati equation (RRE)with Markov modulated switching. By formulating the RRE as a random dynamicalsystem, it is shown that the network achieves weak consensus, i.e., theconditional estimation error covariance at a randomly selected sensor convergesweakly (in distribution) to a unique invariant measure. Further, it is provedthat as $\overline{\gamma} \rightarrow \infty$ this invariant measure satisfiesthe Large Deviation (LD) upper and lower bounds, implying that this measureconverges exponentially fast (in probability) to the Dirac measure$\delta_{P^*}$, where $P^*$ is the stable error covariance of the centralized(Kalman) filtering setup. The LD results answer a fundamental question on howto quantify the rate at which the distributed scheme approaches the centralizedperformance as the inter-sensor communication rate increases.

Distributed Kalman Filtering over Massive Data Sets: Analysis Through Large Deviations of Random Riccati Equations