Prokhorov Metric Convergence of the Partial Sum Process for Reconstructed Functional Data

Motivated by applications in functional data analysis, we study the partial sum process of sparsely observed, random functions. A key novelty of our analysis are bounds for the distributional distance between the limit Brownian motion and the entire partial sum process in the function space. To measure the distance between distributions, we employ the Prokhorov and Wasserstein metrics. We show that these bounds have important probabilistic implications, including strong invariance principles and new couplings between the partial sums and their Gaussian limits. Our results are formulated for weakly dependent, nonstationary time series in the Banach space of d-dimensional, continuous functions. Mathematically, our approach rests on a new, two-step proof strategy: First, using entropy bounds from empirical process theory, we replace the function-valued partial sum process by a high-dimensional discretization. Second, using Gaussian approximations for weakly dependent, high-dimensional vectors, we obtain bounds on the distance. As a statistical application of our coupling results, we validate an open-ended monitoring scheme for sparse functional data. Existing probabilistic tools were not appropriate for this task.