Fluctuations of the empirical quantiles of independent Brownian motions

We consider lid 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 - \u3e alpha is an element of (0, 1). This sequence converges in probability to q(t), the alpha-quantile of the law of B(j)(t). We first show convergence in law in C[0, infinity) of F(n) = n(1/2) (Q(n) - q). We then investigate properties of the limit process F, including its local covariance structure, and Holder-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. (C) 2010 Elsevier B.V. All rights reserved