0. Let **U _{1}**,

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

We think that the liminf expression would be a.s. infinity if
**1/(n b _{n}^{2})** is summable for

2. Let **Y _{n}** be the a.s. unique location of the
maximum of the empirical process

**Question 2.** Find a characterization of
**(b _{n})** such that
liminf

3. Let **Z _{n}** be the total time spent in

**Question 3.** Find a characterization of
**(b _{n})** such that
liminf

4. So far, we are only able to answer these questions for the
particular sequence **b _{n}** =

5. Comments/remarks/solutions welcome.

csaki@math-inst.hu | |

6. This page is jointly maintained with Zhan
Shi.

Last updated : Nov 2, 1999.