Upper and lower class separating sequences for brownian motion with random argument

Let X = X1,X2, . . . be a sequence of random variables, let W be a Brownian motion independent of X and let Zk = W(Xk). A numerical sequence (tk) will be called an upper (lower) class sequence for {Zk} if P(Zk > tk for infinitely many k) = 0 (or 1, respectively). At a first look one might be tempted to believe that a “separating line” (t0k), say, between the upper and lower class sequences for {Zk} is directly related to the corresponding counterpart (s0k) for the process {Xk}. For example, by using the law of the iterated logarithm for the Wiener process a functional relationship t0k = √2s0k log log s0k 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}.