Z-theorems: limits of stochastic equations

Let f(n)(theta, omega) be a sequence of stochastic processes which converge weakly to a limit process f(0)(theta, omega). We show under some assumptions the: weak inclusion of the solution sets theta(n)(omega)= {theta: f(n)(theta, omega) = 0} in the limiting solution set theta(0)(omega) = {theta: f(0)(theta, omega) = 0}. If the limiting solutions are almost surely singletons, then weak convergence holds. Results of this type are called Z-theorems (zero-theorems). Moreover, we give various more specific convergence results, which have applications for stochastic equations, statistical estimation and stochastic optimization.