Z-theorems: limits of stochastic equations

Abstract

Let f_n(θ,ω) be a sequence of stochastic processes which converge weakly to a limit process f₀(θ,ω). We show under some assumptions the weak inclusion of the solution sets ${\theta }_n\left(\omega \right)=\{\theta :f_n(\theta ,\omega )=0\}$ in the limiting solution set ${\theta }_0\left(\omega \right)=\{\theta :f_0(\theta ,\omega )=0\}$ . 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.