We derive Berry-Esséen bounds for both the least squares estimator and the moment estimator of the drift coefficient in Ornstein-Uhlenbeck processes driven by a general Gaussian process G with covariance function R(t, s). The second-order mixed partial derivative of the function R(t, s) can be decomposed into two components: the first component corresponds to the case of fractional Brownian motion (fBm) with Hurst parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_446_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$H\in (0,\frac{1}{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the second component is bounded by the function <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_446_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$(ts)^{H-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>H</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> or a function of bounded variation. This framework encompasses a wide range of well-known fractional Gaussian processes. The derived Berry-Esséen bounds are crucial in enabling the central limit theorems for both estimators. The proofs of the main results are based on an inner product formula that captures the differences between fBm and a general fractional Gaussian process.

Berry-Esséen bounds for the statistical estimators of an Ornstein-Uhlenbeck process driven by a general Gaussian noise

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2025年9月23日

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Yong Chen,
Ying Li,
Hongjuan Zhou

摘要

We derive Berry-Esséen bounds for both the least squares estimator and the moment estimator of the drift coefficient in Ornstein-Uhlenbeck processes driven by a general Gaussian process G with covariance function R(t, s). The second-order mixed partial derivative of the function R(t, s) can be decomposed into two components: the first component corresponds to the case of fractional Brownian motion (fBm) with Hurst parameter $H\in (0,\frac{1}{2})$ $H \in (0, \frac{1}{2})$ and the second component is bounded by the function $(ts)^{H-1}$ ${(t s)}^{H - 1}$ or a function of bounded variation. This framework encompasses a wide range of well-known fractional Gaussian processes. The derived Berry-Esséen bounds are crucial in enabling the central limit theorems for both estimators. The proofs of the main results are based on an inner product formula that captures the differences between fBm and a general fractional Gaussian process.