Gaussian approximation of the empirical process under random entropy conditions

We obtain rates of strong approximation of the empirical process indexed by functions by a Brownian bridge under only random entropy conditions. The results of Berthet and Mason [P. Berthet, D.M. Mason, Revisiting two strong approximation results of Dudley and Philipp, in: High Dimensional Probability, in: IMS Lecture Notes-Monograph Series, vol. 51, 2006, pp. 155–172] under bracketing entropy are extended by combining their method to properties of the empirical entropy. Our results show that one can improve the universal rate v n = o ( log log n ) from Dudley and Philipp [R.M. Dudley, W. Philipp, Invariance principles for sums of Banach space valued random elements and empirical processes, Z. Wahrsch. Verw. Gebiete 62 (1983) 509–552] into v n → 0 at a logarithmic rate, under a weak random entropy assumption which is close to necessary. As an application the results of Koltchinskii [V.I. Kolchinskii, Komlós–Major–Tusnády approximation for the general empirical process and Haar expansions of classes of functions, J. Theoret. Probab. 7 (1994) 73–118] are revisited when the conditions coming in addition to random entropy are relaxed.