We derive sharp finite sample estimates and exact almost sure limit results for local deviations of multivariate empirical processes. These are useful for obtaining, e.g., exact convergence rates of multivariate kernel density estimators. It is also indicated how local properties of multivariate empirical processes may be used to study various problems in nonparametric multivariate analysis.
"The Oscillation Behavior of Empirical Processes: The Multivariate Case." Ann. Probab. 12 (2) 361 - 379, May, 1984. https://doi.org/10.1214/aop/1176993295