# Berry-Ess\'een bounds for parameter estimation of general Gaussian processes

## TL;DR

This paper establishes Berry-Esséen bounds for the convergence rates in the central limit theorem for sums of squares of general Gaussian sequences, without assuming stationarity, using Wiener space analysis and the Optimal Fourth Moment Theorem.

## Contribution

It provides explicit bounds on convergence rates for Gaussian sequences with long memory, extending CLT results to non-stationary, non-self-similar processes like fractional Ornstein-Uhlenbeck.

## Key findings

- Derived explicit Berry-Esséen bounds for Gaussian sequences.
- Applied bounds to parameter estimation in non-stationary Gaussian processes.
- Quantified the trade-off between bias, variance, and memory in estimators.

## Abstract

We study rates of convergence in central limit theorems for the partial sum of squares of general Gaussian sequences, using tools from analysis on Wiener space. No assumption of stationarity, asymptotically or otherwise, is made. The main theoretical tool is the so-called Optimal Fourth Moment Theorem \cite{NP2015}, which provides a sharp quantitative estimate of the total variation distance on Wiener chaos to the normal law. The only assumptions made on the sequence are the existence of an asymptotic variance, that a least-squares-type estimator for this variance parameter has a bias and a variance which can be controlled, and that the sequence's auto-correlation function, which may exhibit long memory, has a no-worse memory than that of fractional Brownian motion with Hurst parameter }$H<3/4$.{\ \ Our main result is explicit, exhibiting the trade-off between bias, variance, and memory. We apply our result to study drift parameter estimation problems for subfractional Ornstein-Uhlenbeck and bifractional Ornstein-Uhlenbeck processes with fixed-time-step observations. These are processes which fail to be stationary or self-similar, but for which detailed calculations result in explicit formulas for the estimators' asymptotic normality.