Let fn(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 $\boldsymbol{\theta}_n(\omega) = \{ \theta : f_n(\theta,\omega) = 0 \}$ in the limiting solution set $\boldsymbol{\theta}_0(\omega) = \{ \theta : f_0(\theta,\omega) = 0 \}$ . 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.