  Motivated by multi-user optimization problems and non-cooperative Nash games
in uncertain regimes, we consider stochastic Cartesian variational inequalities
(SCVI) where the set is given as the Cartesian product of a collection of
component sets. First, we consider the case where the number of the component
sets is large. For solving this type of problems, the classical stochastic
approximation methods and their prox generalizations are computationally
inefficient as each iteration becomes very costly. To address this challenge,
we develop a randomized block stochastic mirror-prox (B-SMP) algorithm, where
at each iteration only a randomly selected block coordinate of the solution is
updated through implementing two consecutive projection steps. Under standard
assumptions on the problem and settings of the algorithm, we show that when the
mapping is strictly pseudo-monotone, the algorithm generates a sequence of
iterates that converges to the solution of the problem almost surely. To derive
rate statements, we assume that the maps are strongly pseudo-monotone and
obtain {a non-asymptotic mean squared error
$\mathcal{O}\left(\frac{d}{k}\right)$, where $k$ is the iteration number and
$d$ is the number of component sets. Second, we consider large-scale stochastic
optimization problems with convex objectives. For this class of problems, we
develop a new averaging scheme for the B-SMP algorithm. Unlike the classical
averaging stochastic mirror-prox (SMP) method where a decreasing set of weights
for the averaging sequence is used, here we consider a different set of weights
that are characterized in terms of the stepsizes and a {parameter}. We show
that using such weights, the objective values of the averaged sequence
converges to the optimal value in the mean sense with the rate
$\mathcal{O}\left(\frac{\sqrt{d}}{\sqrt{k}}\right)$.
