  This paper studies linear stochastic approximation (SA) algorithms and their
application to multi-agent systems in engineering and sociology. As main
contribution, we provide necessary and sufficient conditions for convergence of
linear SA algorithms to a deterministic or random final vector. We also
characterize the system convergence rate, when the system is convergent.
Moreover, differing from non-negative gain functions in traditional SA
algorithms, this paper considers also the case when the gain functions are
allowed to take arbitrary real numbers. Using our general treatment, we provide
necessary and sufficient conditions to reach consensus and group consensus for
first-order discrete-time multi-agent system over random signed networks and
with state-dependent noise. Finally, we extend our results to the setting of
multi-dimensional linear SA algorithms and characterize the behavior of the
multi-dimensional Friedkin-Johnsen model over random interaction networks.
