Coupling inequalities for suprema of non-centered empirical and bootstrap processes

We obtain strong approximations (couplings) to suprema of non-centered empirical processes indexed by possibly unbounded VC-type classes of functions, by suprema of the corresponding Gaussian and bootstrap processes. The bounds on the quality of the couplings are non-asymptotic, which allow us to work with classes of functions whose complexity increases with the sample size. The couplings are not of the Hungarian type and are instead based on the Slepian-Stein methods and Gaussian comparison inequalities. The increasing complexity of function classes and non-centrality of the processes make the results useful for applications in modern nonparametric statistics (Gin\'{e} and Nickl, 2015), in particular allowing to study the power properties of nonparametric tests using Gaussian approximations and the bootstrap.