Estimates for the rate of strong approximation in Hilbert space

TL;DR

This paper extends previous results to provide estimates on the rate of strong Gaussian approximation for sums of i.i.d. Hilbert space-valued random vectors, highlighting the influence of eigenvalue decay on approximation accuracy.

Contribution

It demonstrates how finite-dimensional strong approximation results can be applied to infinite-dimensional Hilbert spaces, emphasizing the role of covariance eigenvalue decay.

Findings

Rate of approximation depends on eigenvalue decay

Finite moments influence approximation quality

Provides estimates for strong Gaussian approximation in Hilbert spaces

Abstract

The aim of this paper is to investigate, which infinite dimensional consequences follow from the main results of recently published paper of the authors (2009) (see Theorems 2 and 3). We show that the finite dimensional Theorem 3 implies meaningful estimates for the rate of strong Gaussian approximation of sums of i.i.d. Hilbert space valued random vectors $\xi_j$ with finite moments $E |\xi_j|^\gamma$, $\gamma>2$. We show that the rate of approximation depends substantially on the rate of decay of the sequence of eigenvalues of the covariance operator of summands.