Asymptotic analysis in multivariate average case approximation with Gaussian kernels

Abstract

We consider tensor product random fields $Y_d$, $d\in \mathbb{N}$, whose covariance funtions are Gaussian kernels. The average case approximation complexity $n^{Y_d}(\epsilon )$ 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 $\epsilon \in (0,1)$. We investigate the growth of $n^{Y_d}(\epsilon )$ for arbitrary fixed $\epsilon \in (0,1)$ and $d\to \infty$. Namely, we find criteria of boundedness for $n^{Y_d}(\epsilon )$ on $d$ and of tending $n^{Y_d}(\epsilon )\to \infty$, $d\to \infty$, for any fixed $\epsilon \in (0,1)$. In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics \begin{eqnarray*} \ln n^{Y_d}(\varepsilon)= a_d+q(\varepsilon)b_d+o(b_d),\quad d\to\infty, \end{eqnarray*} with any $\epsilon \in (0,1)$. Here $q:(0,1)\to \mathbb{R}$ is a non-decreasing function, $(a_d{)}_{d\in \mathbb{N}}$ is a sequence and $(b_d{)}_{d\in \mathbb{N}}$ is a positive sequence such that $b_d\to \infty$, $d\to \infty$. We show that only special quantiles of self-decomposable distribution functions appear as functions $q$ in a given asymptotics. These general results apply to $n^{Y_d}(\epsilon )$ under particular assumptions on the length scale parameters.