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VOLUMES
43.2 43.1 42.2 42.1 41.2 41.1 40.2
40.1 39.2 39.1 38.2 38.1 37.2 37.1
36.2 36.1 35.2 35.1 34.2 34.1 33.2
33.1 32.2 32.1 31.2 31.1 30.2 30.1
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4.1 3.2 3.1 2.2 2.1 1.2 1.1
 WROCŁAW UNIVERSITY OF SCIENCE AND TECHNOLOGY

Contents of PMS, Vol. 31, Fasc. 2,
pages 183 - 202

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.