P. Deheuvels et Dm. Mason, FUNCTIONAL LAWS OF THE ITERATED LOGARITHM FOR LOCAL EMPIRICAL PROCESSES INDEXED BY SETS, Annals of probability, 22(3), 1994, pp. 1619-1661
We introduce the notion of a multivariate local (and tail) empirical p
rocess indexed by sets and establish a number of functional laws of th
e iterated logarithm for such processes. This leads to a unified metho
dology to study the almost sure behavior of various statistics which a
re local functionals of the empirical distribution. Such statistics in
clude density estimators and the Bahadur-Kiefer representation.