On Convergence of Moments for Approximating Processes and Applications to Surrogate Models

We study critera for a pair $(\{ X_n \}$, $\{ Y_n \})$ of approximating processes which guarantee closeness of moments by generalizing known results for the special case that $Y_n = Y$ for all $n$ and $X_n$ converges to $Y$ in probability. This problem especially arises when working with surrogate models, e.g. to enrich observed data by simulated data, where the surrogates $Y_n$'s are constructed to justify that they approximate the $X_n$'s. The results of this paper deal with sequences of random variables. Since this framework does not cover many applications where surrogate models such as deep neural networks are used to approximate more general stochastic processes, we extend the results to the more general framework of random fields of stochastic processes. This framework especially covers image data and sequences of images. We show that uniform integrability is sufficient, and this holds even for the case of processes provided they satisfy a weak stationarity condition.