Rate of convergence for Hilbert space valued processes

TL;DR

This paper establishes optimal convergence rates for stationary Hilbert space valued processes, including linear and certain non-linear sequences, under various dependence conditions, extending classical limit theorems.

Contribution

It provides Berry-Essen type results with sharp rates for both linear and non-linear Hilbert space processes under different dependence structures.

Findings

Optimal convergence rate (n/m)^{1/2} for m-dependent sequences

Rate (n/log n)^{1/2} for weakly geometrically dependent sequences

Extension of Berry-Essen results to Hilbert space valued processes

Abstract

Consider a stationary, linear Hilbert space valued process. We establish Berry-Essen type results with optimal convergence rates under sharp dependence conditions on the underlying coefficient sequence of the linear operators. The case of non-linear Bernoulli-shift sequences is also considered. If the sequence is $m$-dependent, the optimal rate $(n/m)^{1/2}$ is reached. If the sequence is weakly geometrically dependent, the rate $(n/\log n)^{1/2}$ is obtained.