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

TL;DR

This paper establishes criteria for the convergence of moments in approximating stochastic processes, extending classical results to random fields like images, with implications for surrogate modeling and deep neural network applications.

Contribution

It generalizes moment convergence criteria from processes to random fields, including images, and demonstrates uniform integrability as a key condition, even under weak stationarity.

Findings

Uniform integrability ensures moment convergence for random fields.

Extension of classical results to image data and stochastic process sequences.

Applicability to surrogate models and deep neural network approximations.

Abstract

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.