  This paper considers inference over distributed linear Gaussian models using
factor graphs and Gaussian belief propagation (BP). The distributed inference
algorithm involves only local computation of the information matrix and of the
mean vector, and message passing between neighbors. Under broad conditions, it
is shown that the message information matrix converges to a unique positive
definite limit matrix for arbitrary positive semidefinite initialization, and
it approaches an arbitrarily small neighborhood of this limit matrix at a
doubly exponential rate. A necessary and sufficient convergence condition for
the belief mean vector to converge to the optimal centralized estimator is
provided under the assumption that the message information matrix is
initialized as a positive semidefinite matrix. Further, it is shown that
Gaussian BP always converges when the underlying factor graph is given by the
union of a forest and a single loop. The proposed convergence condition in the
setup of distributed linear Gaussian models is shown to be strictly weaker than
other existing convergence conditions and requirements, including the Gaussian
Markov random field based walk-summability condition, and applicable to a large
class of scenarios.
