  The problems discussed in this paper are motivated by general ratio consensus
algorithms, introduced by Kempe, Dobra, and Gehrke (2003) in a simple form as
the push-sum algorithm, later extended by B\'en\'ezit et al. (2010) under the
name weighted gossip algorithm. We consider a communication protocol described
by a strictly stationary, ergodic, sequentially primitive sequence of
non-negative matrices, applied iteratively to a pair of fixed initial vectors,
the components of which are called values and weights defined at the nodes of a
network. The subject of ratio consensus problems is to study the asymptotic
properties of ratios of values and weights at each node, expecting convergence
to the same limit for all nodes. The main results of the paper provide upper
bounds for the rate of the almost sure exponential convergence in terms of the
spectral gap associated with the given sequence of random matrices. It will be
shown that these upper bounds are sharp. Our results complement previous
results of Picci and Taylor (2013) and Iutzeler, Ciblat and Hachem (2013).
