GAUSSIAN APPROXIMATIONS FOR DEPENDENT DATA

This thesis develops sharp Gaussian approximation theory for dependent data, with an emphasis on non-stationary time series and modern machine learning algorithms. The first part establishes strong invariance principles and explicit Gaussian constructions for a broad classes of stationary and non-stationary processes, together with applications to change-point detection and inference, simultaneous confidence bands, and wavelet-based inference. The second part extends this viewpoint to iterative learning systems, deriving refined Gaussian and asymptotic approximations for decentralized federated learning and Q-learning, which exhibit particularly long-range non-stationarity. Across these settings, the thesis combines optimal or near-optimal theoretical rates with extensive numerical experiments, showing how strong approximation methods can serve as a practical foundation for inference in modern applications.