  We show the convergence of the zero relaxation limit in systems of $2 \times
2$ hyperbolic conservation laws with stochastic initial data. Precisely,
solutions converge to a solution of the local equilibrium approximation as the
relaxation time tends to zero. The initial data are assumed to depend on
finitely many random variables, and the convergence is then proved via the
appropriate analogues of the compensated compactness methods used in treating
the deterministic case. We also demonstrate the validity of this limit in the
case of the semi-linear $p$-system; the well-posedness of both the system and
its equilibrium approximation are proved, and the convergence is shown with no
a priori conditions on solutions. This model serves as a prototype for
understanding how asymptotic approximations can be used as control variates for
hyperbolic balance laws with uncertainty.
