Approximate Solutions of Nonlinear Random Operator Equations: Convergence in Distribution

For nonlinear random operator equations where the distributions of the stochastic inputs are approximated by sequences of random variables converging in distribution and where the underlying deterministic equations are simultaneously approximated, we prove a result about tightness and convergence in distribution of the approximate solutions. We apply our result to a random differential equation under Peano conditions and to a random Hammerstein integral equation and its quadrature approximations.