Probability and Mathematical Statistics: Vol. 31, Fasc. 2

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

Christoph Aistleitner
Siegfried Hörmann

Abstract: Let $X = X ,X ,... 1 2$ be a sequence of random variables, let $W$ be a Brownian motion independent of $X$ and let $Z = W (X ) k k$ . A numerical sequence $(t) k$ will be called an upper (lower) class sequence for $(Z ) k$ if

P (Zk > tk forinfinitelymany k) = 0(or 1,respectively).

At a first look one might be tempted to believe that a “separating line” $(t0) k$ , say, between the upper and lower class sequences for $(Zk)$ is directly related to the corresponding counterpart $(s0) k$ for the process $(Xk)$ . For example, by using the law of the iterated logarithm for the Wiener process a functional relationship

eq1

(0.1)

seems to be natural. If $Xk = |W2 (k)|$ for a second Brownian motion $W2$ then we are dealing with an iterated Brownian motion, and it is known that the multiplicative constant $√2-$ in (0.1) needs to be replaced by $2 ⋅3-3∕4$ , contradicting this simple argument.

We will study this phenomenon from a different angle by letting $(Xk)$ be an i.i.d. sequence. It turns out that the relationship between the separating sequences $(s0k)$ and $(t0k)$ in the above sense depends in an interesting way on the extreme value behavior of $(Xk )$ .

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

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