  In this paper, a distributed stochastic approximation algorithm is studied.
Applications of such algorithms include decentralized estimation, optimization,
control or computing. The algorithm consists in two steps: a local step, where
each node in a network updates a local estimate using a stochastic
approximation algorithm with decreasing step size, and a gossip step, where a
node computes a local weighted average between its estimates and those of its
neighbors. Convergence of the estimates toward a consensus is established under
weak assumptions. The approach relies on two main ingredients: the existence of
a Lyapunov function for the mean field in the agreement subspace, and a
contraction property of the random matrices of weights in the subspace
orthogonal to the agreement subspace. A second order analysis of the algorithm
is also performed under the form of a Central Limit Theorem. The
Polyak-averaged version of the algorithm is also considered.
