Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, B j ( t ) , where B j ( 0 ) has a rapidly decreasing, smooth density function f . The empirical quantiles, or pointwise order statistics, are denoted by B j : n ( t ) , and we consider a sequence Q n ( t ) = B j ( 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 B j ( t ) . We first show convergence in law in C [ 0 , ∞ ) of F n = n 1 / 2 ( Q n − 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 .