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

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

Question 1. What is the lower asymptotic behaviour of

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

We think that the liminf expression would be a.s. infinity if 1/(n b_n²) is summable for n, and 0 otherwise.

2. Let Y_n be the a.s. unique location of the maximum of the empirical process G_n. That is, G_n(Y_n) = X_n.

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

3. Let Z_n be the total time spent in R₊ by the empirical process G_n.

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

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

5. Comments/remarks/solutions welcome.