Breuer-Major-Donsker invariance principle

Abstract

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.