Asymptotic analysis of average case approximation complexity of Hilbert space valued random elements

TL;DR

This paper analyzes the asymptotic behavior of the average case approximation complexity for high-dimensional Hilbert space valued random elements, providing criteria for boundedness and growth, and characterizing the asymptotics via probability distribution quantiles.

Contribution

It establishes new criteria for the boundedness and asymptotic growth of approximation complexity in high dimensions, with explicit asymptotic formulas involving probability distribution quantiles.

Findings

Criteria for boundedness of approximation complexity

Asymptotic formulas involving quantiles of stable distributions

Application to tensor products of Euler processes and eigenvalue decay

Abstract

We study approximation properties of sequences of centered random elements $X_d$, $d\in\mathbb N$, with values in separable Hilbert spaces. We focus on sequences of tensor product-type and, in particular, degree-type random elements, which have covariance operators of corresponding tensor form. The average case approximation complexity $n^{X_d}(\varepsilon)$ is defined as the minimal number of continuous linear functionals that is needed to approximate $X_d$ with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. In the paper we investigate $n^{X_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$. Namely, we find criteria of (un)boundedness for $n^{X_d}(\varepsilon)$ on $d$ and of tending $n^{X_d}(\varepsilon)\to\infty$, $d\to\infty$, for any fixed $\varepsilon\in(0,1)$. In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics \begin{eqnarray*} \ln n^{X_d}(\varepsilon)= a_d+q(\varepsilon)b_d+o(b_d),\quad d\to\infty, \end{eqnarray*} at continuity points of a non-decreasing function $q\colon (0,1)\to\mathbb R$. Here $(a_d)_{d\in\mathbb N}$ is a sequence and $(b_d)_{d\in\mathbb N}$ is a positive sequence such that $b_d\to\infty$, $d\to\infty$. Under rather weak assumptions, we show that for tensor product-type random elements only special quantiles of self-decomposable or, in particular, stable (for tensor degrees) probability distributions appear as functions $q$ in the asymptotics. We apply our results to the tensor products of the Euler integrated processes with a given variation of smoothness parameters and to the tensor degrees of random elements with regularly varying eigenvalues of covariance operator.