STRONG APPROXIMATIONS FOR EMPIRICAL PROCESSES INDEXED BY LIPSCHITZ FUNCTIONS

This paper presents new uniform Gaussian strong approximations for empirical processes indexed by classes of functions based on d-variate random vectors (d ≥ 1). First, a uniform Gaussian strong approximation is established for general empirical processes indexed by possibly Lipschitz functions, improving on previous results in the literature. In the setting considered by Rio (Probab. Theory Related Fields 98 (1994) 21–45), and if the function class is Lipschitzian, our result improves the approximation rate n^−1/(2d) to n^{−1/max{d,2}}, up to a polylog(n) term, where n denotes the sample size. Remarkably, we establish a valid uniform Gaussian strong approximation at the rate n^−1/2 log n for d = 2, which was previously known to be valid only for univariate (d = 1) empirical processes via the celebrated Hungarian construction (Komlós, Major and Tusnády, Z. Wahrsch. Verw. Gebiete 32 (1975) 111–131). Second, a uniform Gaussian strong approximation is established for multiplicative separable empirical processes indexed by possibly Lipschitz functions, which addresses some outstanding problems in the literature (Chernozhukov, Chetverikov and Kato, Ann. Statist. 42 (2014) 1564–1597, Section 3). Finally, two other uniform Gaussian strong approximation results are presented when the function class is a sequence of Haar basis based on quasi-uniform partitions. Applications to nonparametric density and regression estimation are discussed.