Z-theorems: Limits of stochastic equations (Q1590232)

For any integer \(n\geq 1\) let \(f_n(\theta, \omega)\) be a continuous (in \(\theta)\) random function defined on \(\Theta\times \Omega_n\) with values in \(\mathbb{R}^m\), \(m\geq 1\), where \(\Theta\) is a given open set in \(\mathbb{R}^d\), \(d\geq 1\), and \((\Omega_n, {\mathcal K}_n,P_n)\) is a probability space. Assume that \(f_n(\theta, \cdot)\) converges weakly as \(n\to\infty\) to a limit random function \(f_0(\theta, \cdot)\) defined on \(\Theta\times \Omega_0\), where \((\Omega_0,{\mathcal K}_0, P_0)\) is a probability space. Under suitable assumptions, the weak inclusion of the solution sets \(\underline\theta_n(\omega)= \{\theta: f_n(\theta, \omega) =0\}\) in the limiting solution set \(\underline\theta_0 (\omega)= \{\theta: f_0 (\theta, \omega)= 0\}\) is proved. If the limiting solutions are almost surely singletons, then weak convergence holds. Results of this type are called Z-theorems (zero-theorems); for a special case see \textit{A. W. van der Vaart} [Stat. Neerl. 49, No. 1, 9-30 (1995; Zbl 0830.62029)]. The authors also give various more specific convergence results, which have applications to stochastic equations, statistical estimation, and stochastic optimization.