Convergence of discrete-stochastic numerical procedures with independent or weakly dependent estimators at grid nodes (Q1973296)

The conjugate wandering method and the polygon of frequencies method are considered to approximate the solution of deterministic integral equations of the second kind \[ \phi (x) = \int_X K(y,x) \phi (y) dy + \psi (x) \] on a compact subset \(D\) of a bounded range \(X \subseteq R^l\), where \(\phi, \psi \in B\) (an appropriate Banach space of integrable functions). In contrast to independent Monte Carlo estimators, the polygon of frequencies method provides weakly dependent estimators for the solution \(\phi\). The probabilistic \(C\)-metric given by \[ \text{Prob} \Bigl\{\sup_{x \in D} |\phi (x) - L_{(M)} \widetilde{\phi}(x)|<G(M,\overline{n},\overline{p}) \Bigr\}> 1 - \varepsilon \] for small \(\varepsilon> 0\) and approximation \(L_{(M)} \widetilde{\phi}(x)\) of transformed solution \(\widetilde{\phi}\), along with convergence in probability, are used to establish convergence of the mentioned numerical approximations while \(G(.) \to + \infty\) when the number of grid points \(M\) and sample points \(\overline{n} \to + \infty\).