NONSTANDARD LOCAL EMPIRICAL PROCESSES INDEXED BY SETS

Let mu(n) denote the empirical measure based on the first n observations from a sequence of independent random variables with common distribution mu in R(d). Let C be a class of Borel subsets of R(d). For any t/-->epsilon R(d) and sequence k(n) with (loglogn)(-1)k(n)-->c epsilon(O, infinity), let theta(n)(C)=k(n)(-1)mu(n)(t/-->+(n(-1)k(n))C-1/d), C epsilon C, be the empirical process at t/--> indexed by C. Under suitable assumptions on mu and C, we obtain functional laws of the iterated logarithm for {theta(n)(C), C epsilon C}.