AI 中文总结
研究平稳高斯序列的布勒尔 - 梅杰 - 唐斯克不变性原理，核心方法是基于可预测鞅分解并系统运用非确定性，贡献是去除相关额外矩假设，给出该原理的新证明。
AI 中文摘要
我们在测试函数的自然有限方差假设下，证明了平稳高斯序列的布勒尔 - 梅杰型唐斯克不变性原理。此结果，即布勒尔 - 梅杰 - 唐斯克原理（BMD 原理），去除了努尔丁和努阿尔特在泛函布勒尔 - 梅杰定理中所施加的额外矩假设。我们的方法不依赖于努尔丁和努阿尔特使用的马利瓦因微积分估计，特别是迈耶不等式。它基于部分和过程的可预测鞅分解，该分解具有独立研究价值。我们还系统地运用了高斯预测理论中的核心概念非确定性。在非确定性情况下，鞅部分由鞅泛函中心极限定理处理，可预测余项通过奥恩斯坦 - 乌伦贝克平滑获得二阶以上的可积性。在确定性情况下，鞅部分消失，且沿整个序列平滑机制不再可用。然而，在协方差函数的一个额外温和假设下，适当的抽取恢复非确定性并将证明简化为非确定性情况。
英文摘要
We prove a Breuer-Major-type Donsker's invariance principle for stationary Gaussian sequences under the natural \emph{finite-variance assumption} on the test function. This result, which we call the \emph{Breuer-Major-Donsker} principle, or simply the \emph{BMD principle}, removes the additional moment assumption imposed in the functional Breuer-Major theorem of Nourdin and Nualart (\emph{Probab. Theory Related Fields}, 2020). Our method does not rely on the Malliavin-calculus estimates used by Nourdin and Nualart, in particular Meyer's inequality. Instead, it is based on a predictable-martingale decomposition of the partial-sum process, which is of independent interest. We also make systematic use of \emph{non-determinism}, a central notion in Gaussian prediction theory. In the non-deterministic case, the martingale part is handled by the martingale functional central limit theorem, while the predictable remainder gains integrability above order two through Ornstein-Uhlenbeck smoothing. In the deterministic case, the martingale part vanishes, and the smoothing mechanism is no longer available along the full sequence. Nevertheless, under an additional mild assumption on the covariance function, a suitable decimation recovers non-determinism and reduces the proof to the non-deterministic case.
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