Berry-Esseen bounds of second moment estimators for Gaussian processes observed at high frequency

TL;DR

This paper derives Berry-Esseen bounds for second moment estimators of stationary Gaussian processes observed at high frequency, establishing their asymptotic normality and applying results to Ornstein-Uhlenbeck processes.

Contribution

It provides the first Berry-Esseen bounds for these estimators, demonstrating their strong consistency and normal approximation in high-frequency settings.

Findings

Strong consistency of the estimators.

Berry-Esseen bounds for the CLT.

Application to Ornstein-Uhlenbeck process parameter estimation.

Abstract

Let $Z:=\{Z_t,t\geq0\}$ be a stationary Gaussian process. We study two estimators of $\mathbb{E}[Z_0^2]$, namely $\widehat{f}_T(Z):= \frac{1}{T} \int_{0}^{T} Z_{t}^{2}dt$, and $\widetilde{f}_n(Z) :=\frac{1}{n} \sum_{i =1}^{n} Z_{t_{i}}^{2}$, where $ t_{i} = i \Delta_{n}$, $ i=0,1,\ldots, n $, $\Delta_{n}\rightarrow 0$ and $ T_{n} := n \Delta_{n}\rightarrow \infty$. We prove that the two estimators are strongly consistent and establish Berry-Esseen bounds for a central limit theorem involving $\widehat{f}_T(Z)$ and $\widetilde{f}_n(Z)$. We apply these results to asymptotically stationary Gaussian processes and estimate the drift parameter for Gaussian Ornstein-Uhlenbeck processes.