Empirical Quantile CLTs For Some Self-Similar Processes

TL;DR

This paper extends CLTs for sample medians from Brownian motions to a broader class of self-similar processes, including fractional Brownian motions and stable processes, with uniform results over quantiles.

Contribution

It generalizes CLTs for sample medians to various self-similar processes and establishes uniform convergence over quantile intervals.

Findings

CLTs hold for fractional Brownian motions and stable processes.

Results are uniform over compact quantile intervals.

Sample function properties are analyzed in connection with CLTs.

Abstract

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.