  Distributed consensus and other linear systems with system stochastic
matrices $W_k$ emerge in various settings, like opinion formation in social
networks, rendezvous of robots, and distributed inference in sensor networks.
The matrices $W_k$ are often random, due to, e.g., random packet dropouts in
wireless sensor networks. Key in analyzing the performance of such systems is
studying convergence of matrix products $W_kW_{k-1}... W_1$. In this paper, we
find the exact exponential rate $I$ for the convergence in probability of the
product of such matrices when time $k$ grows large, under the assumption that
the $W_k$'s are symmetric and independent identically distributed in time.
Further, for commonly used random models like with gossip and link failure, we
show that the rate $I$ is found by solving a min-cut problem and, hence, easily
computable. Finally, we apply our results to optimally allocate the sensors'
transmission power in consensus+innovations distributed detection.
