UPPER AND LOWER CLASS SEPARATING SEQUENCES FOR BROWNIAN
MOTION WITH RANDOM ARGUMENT

Christoph Aistleitner

Siegfried Hörmann

Abstract: Let be a sequence of random variables, let
be a Brownian motion independent of and let .
A numerical sequence will be called an upper (lower) class sequence for
if

At a
first look one might be tempted to believe that a “separating line”

, say, between the
upper and lower class sequences for

is directly related to the corresponding
counterpart

for the process

. For example, by using the law of the iterated
logarithm for the Wiener process a functional relationship

| (0.1) |

seems to be natural. If for a second Brownian motion then we are
dealing with an iterated Brownian motion, and it is known that the multiplicative
constant in (0.1) needs to be replaced by , contradicting this simple
argument.

We will study this phenomenon from a different angle by letting be an
i.i.d. sequence. It turns out that the relationship between the separating sequences and
in the above sense depends in an interesting way on the extreme value behavior of
.

2000 AMS Mathematics Subject Classification: Primary: 60F15; Secondary: 60J65,
60G70.

Keywords and phrases: Brownian motion, extreme values, iterated Brownian motion,
upper-lower class test.