Empirical Quantile Central Limit Theorems for Some Self-Similar Processes

In Swanson (Probab Theory Relat Fields 138:269-304, 2007), a central limit theorem (CLT) for the sample median of independent Brownian motions with value at 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.