Z - theorems : Limits of stochastic equationsVladimir

Let f n (; !) be a sequence of stochastic processes which converge weakly to a limit process f 0 (; !). We show under some assumptions the weak inclusion of the solution sets n (!) = f : f n (; !) = 0g in the limiting solution set 0 (!) = f : f 0 (; !) = 0g. If the limiting solutions are a.s. singletons, then weak convergence holds. Results of his type are called Z-theorems (Zero-theorems). Moreover, we give various more speciic convergence results, which have applications for stochastic equations, statistical estimation and stochastic optimization.