New Berry-Esseen bounds for parameter estimation of Gaussian processes observed at high frequency

The purpose of this paper is to estimate the limiting variance of asymptotically stationary Gaussian processes observed at high frequency, using the second moment estimator (SME). We study rates of convergence of the central limit theorem for the SME in terms of the total variation, Kolmogorov and Wasserstein distances, using some novel techniques and sharp estimates for cumulants. We apply our approach to provide Berry-Esseen bounds in Kolmogorov and Wasserstein distances for estimators of the drift parameters of Gaussian Ornstein-Uhlenbeck processes. Moreover, we prove that most of our estimates are strictly sharper than the ones obtained in the existing literature.

💡 Research Summary

The paper addresses the problem of estimating the limiting variance of a (possibly non‑semimartingale) Gaussian process that is observed at high frequency, i.e., at discrete times t_i = iΔ_n with Δ_n → 0 as the number of observations n → ∞. The authors focus on the second‑moment estimator (SME) defined by the empirical average of squared observations and study the rate at which the normalized estimator satisfies a central limit theorem (CLT).

The main objects are the stationary Gaussian process Z_t and the asymptotically stationary process X_t = Z_t – e^{–θt}Z_0. For each process the SME is v_n(Z) = (1/n)∑{i=0}^{n-1} Z{t_i}^2 and v_n(X) = (1/n)∑{i=0}^{n-1} X{t_i}^2. After centering by the true variance ρ(0)=E