Approximation complexity of homogeneous sums of random processes

TL;DR

This paper investigates how the complexity of approximating sums of identical zero-mean random processes grows with the number of processes, focusing on the average case error and applying results to Wiener process sums.

Contribution

It provides new insights into the approximation complexity of homogeneous sums of random processes, including growth rates and specific applications to Wiener processes.

Findings

Growth of approximation complexity with increasing process number

Explicit results for sums of Wiener processes

Analysis of approximation error thresholds

Abstract

We study approximation properties of additive random fields $Y_d$, $d\in\mathbb{N}$, which are sums of zero-mean random processes with the same continuous covariance functions. The average case approximation complexity $n^{Y_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate $Y_d$, with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. We investigate the growth of $n^{Y_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$. The results are applied to sums of standard Wiener processes.