Questions on empirical processes

0. Let U1, U2, U3, ... be a sequence of independent random variables with the common uniform (0, 1) distribution. For t in the interval [0, 1], let

Fn(t) = #{ i < n+1: Ui < t} ,

which is the empirical distribution function based on the first n observations. Define the empirical process

Gn(t) = n1/2 (Fn(t) - t).

Question 1. What is the lower asymptotic behaviour of

Xn = sup Gn(t) ?

(The symbol "sup" denotes the supremum over all t). More precisely, we ask for a characterization of non-decreasing sequences (bn), such that, almost surely (a.s.), when n goes to infinity, liminf bnXn is positive/zero.

We think that the liminf expression would be a.s. infinity if 1/(n bn2) is summable for n, and 0 otherwise.

2. Let Yn be the a.s. unique location of the maximum of the empirical process Gn. That is, Gn(Yn) = Xn.

Question 2. Find a characterization of (bn) such that liminf bnYn is positive/zero, a.s.

3. Let Zn be the total time spent in R+ by the empirical process Gn.

Question 3. Find a characterization of (bn) such that liminf bnZn is positive/zero, a.s.

4. So far, we are only able to answer these questions for the particular sequence bn = (log n)r, with r < 0.