Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, Bj(t), where Bj(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by Bj:n(t), and we consider a sequence Qn(t)=Bj(n):n(t), where j(n)/n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of Bj(t). We first show convergence in law in C[0,∞) of Fn=n1/2(Qn−q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.