  This paper studies the convergence of the estimation error process and the
characterization of the corresponding invariant measure in distributed Kalman
filtering for potentially unstable and large linear dynamic systems. A gossip
network protocol termed Modified Gossip Interactive Kalman Filtering (M-GIKF)
is proposed, where sensors exchange their filtered states (estimates and error
covariances) and propagate their observations via inter-sensor communications
of rate $\overline{\gamma}$; $\overline{\gamma}$ is defined as the averaged
number of inter-sensor message passages per signal evolution epoch. The
filtered states are interpreted as stochastic particles swapped through local
interaction. The paper shows that the conditional estimation error covariance
sequence at each sensor under M-GIKF evolves as a random Riccati equation (RRE)
with Markov modulated switching. By formulating the RRE as a random dynamical
system, it is shown that the network achieves weak consensus, i.e., the
conditional estimation error covariance at a randomly selected sensor converges
weakly (in distribution) to a unique invariant measure. Further, it is proved
that as $\overline{\gamma} \rightarrow \infty$ this invariant measure satisfies
the Large Deviation (LD) upper and lower bounds, implying that this measure
converges 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 how
to quantify the rate at which the distributed scheme approaches the centralized
performance as the inter-sensor communication rate increases.
