  In a paper of Jason Swanson, a CLT for the sample median of independent
Brownian motions with value 0 at 0 was proved. Here we extend this result in
two ways. We prove such a result for a collection of self-similar processes
which include the fractional Brownian motions, all stationary, independent
increment symmetric stable processes tied down at 0 as well as iterated and
integrated Brownian motions. Second, our results hold uniformly over all
quantiles in a compact sub-interval of (0,1). We also examine sample function
properties connected with these CLTs.
