Z-theorems: Limits of stochastic equations

Abstract

Let fn(è, ù) be a sequence of stochastic processes which converge weakly to a limit process f 0(è, ù). We show under some assumptions the weak inclusion of the solution sets èn(ù) fè : fn(è, ù) 0g in the limiting solution set è0(ù) fè : f 0(è, ù) 0g. 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 speci®c convergence results, which have applications for stochastic equations, statistical estimation and stochastic optimization.