  We consider stochastic variational inequalities with monotone operators
defined as the expected value of a random operator. We assume the feasible set
is the intersection of a large family of convex sets. We propose a method that
combines stochastic approximation with incremental constraint projections
meaning that at each iteration, a step similar to some variant of a
deterministic projection method is taken after the random operator is sampled
and a component of the intersection defining the feasible set is chosen at
random. Such sequential scheme is well suited for applications involving large
data sets, online optimization and distributed learning. First, we assume that
the variational inequality is weak-sharp. We provide asymptotic convergence,
feasibility rate of $O(1/k)$ in terms of the mean squared distance to the
feasible set and solvability rate of $O(1/\sqrt{k})$ (up to first order
logarithmic terms) in terms of the mean distance to the solution set for a
bounded or unbounded feasible set. Then, we assume just monotonicity of the
operator and introduce an explicit iterative Tykhonov regularization to the
method. We consider Cartesian variational inequalities so as to encompass the
distributed solution of stochastic Nash games or multi-agent optimization
problems under a limited coordination. We provide asymptotic convergence,
feasibility rate of $O(1/k)$ in terms of the mean squared distance to the
feasible set and, in the case of a compact set, we provide a near-optimal
solvability convergence rate of $O\left(\frac{k^\delta\ln k}{\sqrt{k}}\right)$
in terms of the mean dual gap-function of the SVI for arbitrarily small
$\delta>0$.
