The paper considers gossip distributed estimation of a (static) distributed
random field (a.k.a., large scale unknown parameter vector) observed by
sparsely interconnected sensors, each of which only observes a small fraction
of the field. We consider linear distributed estimators whose structure
combines the information \emph{flow} among sensors (the \emph{consensus} term
resulting from the local gossiping exchange among sensors when they are able to
communicate) and the information \emph{gathering} measured by the sensors (the
\emph{sensing} or \emph{innovations} term.) This leads to mixed time scale
algorithms--one time scale associated with the consensus and the other with the
innovations. The paper establishes a distributed observability condition
(global observability plus mean connectedness) under which the distributed
estimates are consistent and asymptotically normal. We introduce the
distributed notion equivalent to the (centralized) Fisher information rate,
which is a bound on the mean square error reduction rate of any distributed
estimator; we show that under the appropriate modeling and structural network
communication conditions (gossip protocol) the distributed gossip estimator
attains this distributed Fisher information rate, asymptotically achieving the
performance of the optimal centralized estimator. Finally, we study the
behavior of the distributed gossip estimator when the measurements fade (noise
variance grows) with time; in particular, we consider the maximum rate at which
the noise variance can grow and still the distributed estimator being
consistent, by showing that, as long as the centralized estimator is
consistent, the distributed estimator remains consistent.Comment: Submitted for publication, 30 page
