Approximation of multiparametric Anderson-Darling processes

A. A. Khartov

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 33, Tome 515 (2022), pp. 214-232

Résumé

We consider a sequence of Gaussian random fields that are growing tensor products of generalized Anderson-Darling processes with a given sequence of main parameters $(\mu_j)_{j\in\mathbb{N}}$ that characterize a proximity to the Gaussian white noise. The average case approximation complexity for a given $d$-parametric random field is defined as the minimal number of values of continuous linear functionals that is needed to approximate the field with relative $2$-average error not exceeding a given threshold $\varepsilon$. In the paper we obtain logarithmic asymptotics of the average case approximation complexity for such random fields for fixed $\varepsilon\in(0,1)$ and $d\to\infty$ for in fact homogeneous case $\mu_j\to c$, $j\to\infty$, where $c\in(0,\infty)$ is a constant, and for the case $\mu_j\to\infty$, $j\to\infty$, that is rather non-standard for the practice of the similar approximation problems.